Papers

We consider the Cauchy problem for the nonlinear Dirac equation on a noncompact $N$-star metric graph $G$, \[ \mathrm{i}\partial_t \psi = D\psi - |\psi|^{p-2}\psi, \qquad \psi(0)=\psi_0, \] where $p\ge3$, $\psi:\mathbb{R}\times G\to\mathbb{C}^2$ and $D$ denotes the self-adjoint Dirac-Kirchhoff operator on $G$. Using Bourgain-type spaces defined through the spectral resolution of $D$, together with elementary $L^\infty$ bounds for the Dirac flow and fractional Nemytskii estimates below the trace threshold, we prove local well-posedness for initial data \[ \psi_0\in H_D^s(G)\cap L^\infty(G;\mathbb C^2), \qquad 0\le s<\frac12 . \] The corresponding solution belongs to \[ C([0,T];H_D^s(G))\cap X_T^{s,b}\cap L^\infty([0,T]\times G). \] Moreover, $\|\psi(t)\|_{L^2(G;\mathbb{C}^2)}$ is conserved along the solution on the existence interval. We also establish a blow-up alternative in the combined $H_D^s$ and space-time $L^\infty$ control norm.

A number of the form $a_s+a_d \epsilon$, where $a_s, a_d \in\mathbb{C}^{n \times n}$ and $\epsilon$ is an infinitesimal unit satisfying $\epsilon^2=0$, is called a dual number. A matrix with dual number entries is known as dual matrix. The objective of this article is to derive a polar decomposition of dual matrices and to study the existence and uniqueness conditions. To illustrate these results, some examples are constructed.

We develop a coordinate-anchored pointwise theory of fractional tangent functionals on bounded rectangles. By working on the range of the coordinatewise Riemann-Liouville integral, we define anchored fractional operators by exact inversion and prove a representation theorem: every coordinate fractional tangent functional is a scalar multiple of evaluation of the inverse operator at the base point, and the normalized one is unique. At interior points, the resulting fractional tangent space acts faithfully on the common anchored space. We then construct an exterior algebra over the polynomial algebra generated by the fractional coordinate primitives. Its fractional exterior differential defines a nilpotent algebraic chain complex, admits an explicit polynomial Poincare homotopy, and is related to the ordinary polynomial de Rham complex by a positive diagonal rescaling. Since this fractional differential is not a graded derivation for the ordinary wedge product, the global object is a de Rham-type chain complex. We also prove a rigidity theorem for positive-cone-preserving linear coordinate changes preserving the fractional coframe and formulate a Lorentzian coframe-attached Maxwell-type system, deriving charge conservation and fractional wave equations within the polynomial coefficient class.

The dimension of a poset $P$ is the minimum positive integer $d$ such that $P$ is an induced subposet of $\mathbb{R}^d$ equipped with the product order. We give a constant-factor polynomial-time approximation algorithm for computing dimension in the class of posets with a planar (Hasse) diagram. While computing the dimension of a poset is NP-hard in general, the computational complexity of the problem for planar posets remains open. The algorithmic result is driven by a structural understanding of the canonical obstruction to small dimension: standard examples. A longstanding problem, originating in the early 1980s, asked whether every poset with a planar diagram has dimension bounded by a function of the maximum order of a standard example that it contains. In the first paper of the series, we have resolved the problem in a more general setting of posets with planar cover graphs by establishing a polynomial bound. We prove a stronger bound in the original setting, namely, for every poset $P$ with a planar diagram $\mathrm{dim}(P) \leq 96\mathrm{se}(P)+672$, where $\mathrm{dim}(P)$ denotes the dimension of $P$ and $\mathrm{se}(P)$ denotes the maximum order of a standard example contained in $P$.

In this work, we introduce a fifth-order well-balanced (WB) path-conservative A-WENO scheme with the central-upwind numerical fluxes (PCCU-5) for the Ripa model. The proposed scheme is capable of exactly preserving a variety of steady states, including still-water, moving-water, isobaric, and constant water height ones. This goal is achieved with the help of a flux globalization technique: The source terms are incorporated into the fluxes, resulting in a quasi-conservative system, for which central-upwind numerical fluxes are computed using the path-conservative integration. The proposed A-WENO scheme utilizes a WENO interpolation of the equilibrium variables rather than the conservative ones to ensure the WB property. In addition, we perform the WENO interpolation of the local characteristic equilibrium variables to mitigate numerical oscillations near discontinuities. We perform a series of numerical experiments, which demonstrate that the proposed fifth-order WB PCCU-5 scheme achieves high resolution and clearly outperforms its second-order counterpart. Our numerical results also demonstrate the importance of the local characteristic projection for significantly reducing (eliminating) numerical oscillations near discontinuities.

A vertex-transitive closure of $\Gamma$ is a vertex-transitive supergraph of $\Gamma$ on the same vertex set. The vertex-transitive number of a graph $\Gamma$, denoted by $d_{vt}(\Gamma)$, is the smallest integer for which there exists a $d_{vt}(\Gamma)$-regular vertex-transitive closure of $\Gamma$. In this paper we use various algebraic and combinatorial methods to study vertex-transitive closures of graphs and the associated vertex-transitive number.

A $\{P_2,P_3\}$-factor in a graph $G$ is a factor of $G$ in which every component is a path on two or three vertices. Let $T\Box P_n$ be the Cartesian product of a tree $T$ and a path on $n$ vertices. Kao and Weng proved that $T\Box P_n$ is hamiltonian if $T$ has a path factor and $n$ is a sufficiently large even integer. In this article we prove that, for every odd $n$, there exists a tree $T$ of maximum degree 4 that has a $\{P_2,P_3\}$-factor such that $T\Box P_n$ is not hamiltonian, thereby refuting a conjecture by Kao and Weng.

Contention resolution schemes (CRS) are a fundamental and widely applied tool for rounding fractional solutions subject to combinatorial constraints. However, the known analyses of CRS generally only guarantee lower bounds on the expected value and concentration on the upper tail, but no concentration on the lower tail. Thus, CRS are generally not applicable to problems that contain covering constraints, since certifying a covering constraint holds requires a lower tail bound. Our main contribution is to derive lower tail bounds for the output of a particular contention resolution scheme, the random-order CRS of Adamczyk and Włodarczyk, which we call AW. We show that every linear function of the rounded solution attains a constant fraction of its expectation with a failure probability that is dimension-free, depending only on the expected value and on the number of matroids, but not on the size of the ground set. Our analysis is driven by a new property we call \emph{strong $\lambda$-boundedness}, which strengthens the known $\lambda$-boundedness of AW by providing two-sided control on how rounding propagates between elements. We then introduce a random process capturing AW, a sequential selection process, that may be of independent interest. We prove lower tail bounds for any strongly $\lambda$-bounded sequential selection process. To demonstrate the applicability of our new tail bounds, we apply them to two problems involving covering constraints. The first result is an $O(k \log k)$-approximation for $k$-matroid intersection coloring (improving the prior $O(k^2)$) when the chromatic number of at least one matroid is $\Omega(k^3 \log n)$, where $n$ is the number of elements. The second is the first bicriteria approximation algorithm for monotone submodular maximization under $k$ matroid constraints together with packing and covering constraints.

Let $k$ be an imaginary quadratic field and let $p \ne 2$ be a prime number, split in $k$ into ${\mathfrak p}{\overline {\mathfrak p}}$. We assume that the $p$-class group of $k$ is trivial. Let $\delta \geq 0$ be the ${\mathfrak p}$-valuation of the ${\overline {\mathfrak p}}$-Fermat quotient of the fundamental ${\mathfrak p}$-unit $x$ of $k$. Let $K/k$ be any bi-ramified ${\mathbb{Z}}_p$-extension and let $p^e$ be the degree of the inertia field of ${\mathfrak p}$, $\overline {\mathfrak p}$ being totally ramified. We prove that if $e \geq \delta$, then $\lambda(K/k) = 1$, $\mu(K/k) = 0$; if $e<\delta$ a characterization is obtained from Iwasawa invariants of the $S^{\mathfrak p}$-class groups. This approach only uses generalizations of Chevalley-Herbrand formulas and the non-nullity of a $p$-adic regulator ${\mathcal R}^{\mathfrak p}_\delta$ in incomplete $p$-ramification. It provides effective and computable results that complement some aspects of Iwasawa theory. Conjecture states that only the cyclotomic ${\mathbb{Z}}_p$-extension is ”exceptional”; justifications are given. A pari/gp program computes $\delta$ and ${\mathcal R}^{\mathfrak p}_\delta$, for $p=3$, $e=1$.

Non-binary successive cancellation list (NB-SCL) decoding expands each surviving path into $q$ candidate branches at every information symbol, which causes high path expansion, sorting, and pruning complexity. To address this issue, this paper proposes low-complexity list decoding algorithms for $2\times2$ kernel non-binary polar codes (NBPCs). First, we design a split-reduced non-binary successive cancellation list (SR-NBSCL) decoder that skips path splitting when the current symbol is sufficiently reliable. We then exploit the final Rate-1 node structure and switch the last group of information symbols to simplified non-binary successive cancellation (NB-SC) decoding, resulting in the enhanced split-reduced non-binary successive cancellation list (ESR-NBSCL) decoder. To further reduce branch expansion at unreliable symbols, we introduce an accumulated reliability-deviation (ARD) metric and propose an adaptive branch-pruning non-binary successive cancellation list (ABP-NBSCL) decoder, which prunes unreliable candidate branches before sorting and then reduces the dominant sorting complexity. Simulation results show that the proposed decoders achieve frame-error-rate (FER) performance close to that of conventional NB-SCL decoding with much lower complexity. In particular, the ABP-NBSCL decoder reduces the path splitting number (PSN) by more than $80\%$ at several tested signal-to-noise ratios (SNRs), with a negligible performance loss.

We study matrix feasibility problems "find X in K intersect C" where K is a closed convex matrix set and C = {X : sigma(X) in S} is defined by a convex constraint S on the ordered singular values. Using the classical spectral transfer identity, projection onto C reduces to an SVD plus a small quadratic program. For seven natural polyhedral S we embed this projector in a plain alternating projection (AP) loop. We experiment with two concrete families of K - linear constraints (an affine subspace intersected with an entrywise box) and ellipsoidal constraints (a non-centered anisotropic Frobenius ellipsoid) - although the method applies equally to more general convex constraints. The experiments expose three regimes: rapid feasibility, slow tail convergence, or informative infeasibility plateaus.

We study deformations of polytopes that preserve edge lengths and face coplanarities. This gives rise to a notion of rigidity, for which we develop a second-order theory together with an effective algorithm for testing second-order rigidity symbolically. The strength of these new tools is demonstrated on several polytope classes, including the previously intractable regular dodecahedron, which we find to be rigid. We also find that a single tested polytope evades our techniques and will therefore provide a simple test case for future developments of tools of even higher order. This paper also contains a study of so-called edge-length perturbations. We point out connections between rigidity and the ability to realize polytopes with slightly perturbed edge lengths. To explore these connections in practice, we dedicate a section to another case study of the regular dodecahedron. We investigate the local behavior of its realization space with a view towards edge-length perturbations, singularities and generic global rigidity.

We study a $(1+1)$-dimensional directed polymer in a spatially correlated random environment generated by power-law tail variables: $\omega(i,x)=\sum_{y\in\mathbb Z}\psi_{y-x}\xi(i,y), \psi_y\sim \lambda_r |y|^{-r}, r\in(1/2,1)$, where the variables $\xi(i,y)$ are i.i.d. and have a regularly varying right tail with exponent $\alpha>2$. The spatial covariance of the environment has long-range decay with Hurst parameter $H=\frac32-r\in(1/2,1)$. We identify the limiting fluctuations of the log-partition function in the intermediate disorder regime and show that the critical tail exponent is $\alpha_c=\frac{3}{H}=\frac{6}{3-2r}$. When $\alpha>\alpha_c$, the model has the same scaling limits as the corresponding Gaussian spatially correlated polymer: if $\beta_NN^{H/2}\to\beta\in(0,\infty)$, the centered log-partition function converges to the logarithm of the solution of the stochastic heat equation driven by fractional spatial noise; if $\beta_NN^{H/2}\to0$, its normalized fluctuation converges to a centered Gaussian law. In the regime $2<\alpha\le\alpha_c$, at the scale $\beta_NN^{H/2}=\beta N^{H/2}/l(N^{3/2})$, the log-partition function still satisfies Gaussian fluctuation. The main ingredient is a truncation comparison argument adapted to long-range moving-average environments, together with an invariance principle for polynomial chaos. Due to the non-locality of the environments, we perform a far-near field analysis, as well as multiscale analysis, to prove that the truncated version does not change the log-partition function at the corresponding scales.

In this note, we resolve the remaining open case of a conjecture by Han and Zhao concerning the codegree threshold for Hamilton $\ell$-cycles in $k$-uniform hypergraphs. Specifically, we prove that for integers $k\ge 3$, $3k/4\le \ell<k$, with $k\not\equiv 0 \pmod{k-\ell}$, and for all sufficiently large $n$ divisible by $k-\ell$, every $n$-vertex $k$-uniform hypergraph $H$ satisfying \[ \delta_{k-1}(H)\ge \frac{n}{(k-\ell)\left\lceil \frac{k}{k-\ell}\right\rceil} \] contains a Hamilton $\ell$-cycle. Our proof builds on the framework of Gan, Han and Xu, and refines their argument to obtain, at the exact threshold, the required family of paths.

For a quasi-projection operator $Q_j(f,\phi, \widetilde\phi)$ formed by a compactly supported function $\phi$ and a compactly supported distribution $\widetilde\phi$ with dilation matrix $M$, we establish necessary and sufficient conditions under which it provides prescribed simultaneous approximation and simultaneous density orders for a function $f$ from the Sobolev space $H^s({\mathbb R}^d)$ and for its derivatives. The obtained criteria are then used to endow MRA-based dual wavelet frames in a pair of dual Sobolev spaces with the desired simultaneous approximation properties.

The current state of the art in AI/ML rests on deep neural architectures, which, in general, suffer from a lack of interpretability. Interpretability is crucial to gleaning insights while analyzing experimental data, where quantitative predictions may not be adequate for a scientist. We present a three layer neural architecture, SAMPAT (Smooth Approximation via Multivariate Polynomials and Analytic Transformations), that can provably learn a continuous, everywhere differentiable function, that can approximate any smooth function arbitrarily closely. SAMPAT's approximant can be expressed as a closed and compact algebraic, analytic expression, providing complete interpretability. Experiments on synthetic and benchmark datasets indicate that SAMPAT yields competitive performance with simpler representations. For many tasks, a two layer SAMPAT suffices. By imposing restrictions on the connectivity between neurons, SAMPAT may be used to provide a range of approximants, including regular and trigonometric polynomials, rational expressions, Gaussians, mixtures of Gaussians, as well as arbitrary combinations of the same; without restrictions, it learns a suitable structure. SAMPAT may be used to factorize polynomials and model nonlinear systems. With the addition of skip connections, a 4 to 6 layer SAMPAT is adequate to represent a substantive range of methods widely used in AI/ML, allowing the choice of a model's family, not just its parameters, to also be optimized as part of the learning process.

If $\mathfrak u$ and $\mathfrak v$ are Lie algebras, then the product $\mathrm{Out}(\mathfrak u)\times \mathrm{Out}(\mathfrak v)$ of their outer automorphism groups naturally acts on the set of outer Lie algebra morphisms from $\mathfrak u$ to $\mathfrak v$; the stabilizer of the outer class of a given such morphism is then a subgroup of $\mathrm{Out}(\mathfrak u)\times \mathrm{Out}(\mathfrak v)$. We show that this leads to two related interpretation of the Grothendieck-Teichmüller group $\mathsf{GRT}_1(\mathbf k)$, where $\mathfrak u,\mathfrak v$ are the Lie algebra of infinitesimal braids on the plane (resp. framed infinitesimal braids on the sphere) with 3 and 4 (resp. 4 and 5) strands: namely, it can be expressed as the joint intersection of the stabilizer groups of the outer classes of certain strand doubling morphisms $\phi$ and $\psi$ with $\mathrm{Out}^*(\mathfrak u)\times\mathrm{Out}(\mathfrak v)$, where $\mathrm{Out}^*(\mathfrak u)$ is a subgroup of $\mathrm{Out}(\mathfrak u)$ of outer classes of inertia-preserving automorphisms of $\mathfrak u$.

This paper establishes a central limit theorem (CLT) for functionals of $M$-bounded persistence diagrams arising from germ-grain random set models. Building on stabilisation methods for marked point processes, we show that, under certain conditions, these topological summaries exhibit asymptotic normality as the observation window increases, particularly for models with exponential decay of correlations. These results are applied in goodness-of-fit tests designed to detect spatial interactions such as clustering or repulsion. Using test statistics derived from rectangular partitions of persistence diagrams and functional summaries (e.g., the APF or the support function of the lift zonoid), the study distinguishes between different models. Finally, the methodology is applied to histological images of breast tissue.

We present a general computational framework for macroscopic nonlocal traffic flow models on networks with multiple commodities. The model combines scalar conservation laws on edges with nonlocal velocity functions and couples them via buffer-based junction dynamics. Routing is defined in general as a prescription for distributing drivers across outgoing road segments. As an example, we implement dynamic k-shortest-path routing, where travel times along roads and waiting times at intersections are used to compute shortest paths to the commodities' destinations at each time step, and drivers are distributed accordingly. In another example, we optimize routing over a considered time horizon to minimize the total travel time. This framework naturally creates a feedback loop between traffic evolution and route choice. Numerical examples, ranging from small test cases to large grid-like networks, demonstrate the robustness of the approach and allow for a comparison of different routing strategies.

We consider $0/1$ packing problems $\max\{c^T x \colon Ax \leq 1, \, x \in \{0,1\}^n\}$, with $A \in \mathbb{R}_{\geq 0}^{m \times n}$. A way to solve such problems is via tightening the linear programming relaxation $P$ with Gomory cutting-planes. The Gomory-closure $P'$ of $P$ is the intersection of $P$ with all its cutting planes. The optimization problem over $P'$ is NP-hard. Mastrolilli (2020) has shown that for fixed ${\epsilon}>0$, the Lasserre hierarchy yields a polynomial-size convex but non-polyhedral extended formulation that approximates $P'$ up to a factor of $1+{\epsilon}$. Our main result is the construction of a polyhedral and polynomial extended formulation that approximates $P'$ with the same approximation guarantee. Our construction is based on first principles. Like Mastrolilli's approach, ours also applies to higher iterates $P^{(t)}$ for fixed $t$ and ${\epsilon}>0$. In contrast to an explicit construction, communication complexity provides an alternative way to describe extended formulations. Using this approach we obtain a quasi-polynomial polyhedral extended formulation for the above problem that is superior in some parameter regimes. To achieve this, we describe a communication protocol extending Yannakakis' protocol to decide whether the clique of Alice and the stable set of Bob intersect.

Grant-Free (GF) random access is promising for low Earth orbit satellite Internet due to its reduced access latency. However, existing schemes suffer from poor performance in massive connectivity scenarios. To address this challenge, we firstly propose an iterative residual feedback multi-measurement vector approximate message passing algorithm. This algorithm leverages multi-domain synergistic sparsity in the spatial-frequency and angular-delay domains to alternately perform active user terminal detection (AUD) and channel estimation (CE). Additionally, a residual feedback mechanism is incorporated to suppress error accumulation, thereby enhancing AUD performance. Furthermore, conventional data detection (DD) methods significantly degrade when active user terminals are spatially close or outnumber the satellite's receive antennas, making the demodulation problem rank-deficient or underdetermined. To mitigate this, we design a data modulation scheme via joint spatial-frequency multi-domain spreading, which utilizes observations from both spatial and frequency domains to facilitate multi-domain DD. Simulation results demonstrate that the proposed scheme significantly outperforms existing GF methods in terms of AUD accuracy, CE precision, and bit error rate, especially under conditions of low effective pilot length and practical signal-to-noise ratios.

Symmetric Macdonald polynomials of $N$ variables provide eigenfunctions of the $N$-body trigonometric Ruijsenaars-Schneider integrable system at particular eigenvalues. In order to construct eigenfunctions with arbitrary eigenvalues, M. Noumi and J. Shiraishi used a recursion in $N$ (branching rule) for the symmetric Macdonald polynomials and analytically continued them. This generated a power series, which is a part of triad (universal solution). In the present paper, we demonstrate that a similar procedure is available for another integrable system, $N$-body Cherednik integrable system inspired by the DAHA of type $A$, which has non-symmetric Macdonald polynomials as its polynomial eigenfunctions. However, in this system, the generic eigenfunction is more complicated: it is not just a simple power series as in the Noumi-Shiraishi case, but has an involved structure with $N!$ branches, each of them being a sum over the Weyl chambers of power series of the Noumi-Shiraishi type. As an illustration, we also provide explicit formulas for particular cases.

The study of vacuum is important in understanding compressible flows. In particular, physical vacuum, which ensures a finite, nontrivial pressure-induced acceleration in the normal direction on the moving boundary, naturally arises in the study of the motion of gaseous stars. In this paper, we analyze the free boundary problem for the three-dimensional compressible Navier–Stokes–Poisson equations with degenerate viscosities for self-gravitating viscous gaseous stars. For the spherically symmetric and barotropic motion, we establish the global existence of the unique classical solution without any restriction on the size of the initial data. Our solutions are classical up to the moving boundary for positive times, and preserve the physical vacuum boundary behavior of the Lane–Emden star configuration.

Zak-Orthogonal Time Frequency Space (OTFS) modulation is known to be robust to Doppler spread in high mobility scenarios when compared to Orthogonal Frequency Division Multiplexing (OFDM). This is due to the fact that the channel response to a Zak-OTFS carrier within a frame can be accurately estimated from the channel response to another carrier within the same frame. However, an important open problem and question is whether inter-frame channel prediction is possible with Zak-OTFS, i.e., is it possible to accurately predict the channel response to a Zak-OTFS carrier in a frame based on knowledge of the channel response to some Zak-OTFS carrier in another frame (i.e., not the same frame). In this paper we show that indeed inter-frame channel prediction is possible. We show that the effective DD domain channel filter coefficients vary in a deterministic manner as we move from current to future frames in time and frequency. We also show that the subspace spanned by channel filter coefficients of consecutive frames in time/frequency is invariant to discrete shifts in time and frequency. We exploit the deterministic variation and subspace invariance to propose a novel deterministic ESPIRIT-type method which uses the effective DD domain channel filter taps/coefficients estimated in training frames (i.e., current/past frames in time and frequency having both pilot and data carriers) to predict the effective DD domain channel filter for frames which are several tens of frames in future and several tens of frames away in frequency.

The groundbreaking development of generative artificial intelligence (AI) is rapidly boosting the ability to generate content such as images and videos, reshaping communication paradigms. This article introduces generative communications (GenCom), a novel paradigm for 6G networks in which large AI models (LAMs) drive semantic understanding, reasoning, and content generation, embedding these into the communication process. Unlike traditional systems that strictly pursue accurate bit transmission, GenCom enables transmitters to convey only minimal yet sufficient information, while receivers leverage shared generative priors and knowledge bases to synthesize the intended output. Communication is thus redefined as controlled generation rather than data reproduction. We formalize the concept of GenCom, clarify its AI-native and generation-driven properties, and present its core mechanisms. A two-layer GenCom architecture supported by key enabling technologies is proposed, and analysis of four representative application scenarios demonstrates that GenCom offers ultra-efficient transmission, semantic-level robustness, and new network functions. Finally, we outline future research directions, including foundational theory and real-time processing, highlighting a promising pathway toward 6G networks.

The article studies finite element approximations of a quasilinear elliptic heat-conduction problem with inhomogeneous mixed boundary conditions. The conductivity tensor is matrix-valued, anisotropic, possibly nonsymmetric, and dependent on both position and temperature, rendering the problem nonlinear, nonmonotone, and nonpotential. The nonlinear algebraic system arising from the Galerkin discretization is solved using Newton's method, with a posteriori guarantees provided by a computable Newton-Kantorovich criterion and a mesh-dependent stopping rule that ensures that the algebraic error is asymptotically negligible relative to the discretization error. Since the discrete solution need not be unique, we prove that every discrete solution satisfies the optimal convergence rates. The analysis combines mixed-boundary elliptic regularity with an Aubin-Nitsche duality argument adapted to the quasilinear setting. Numerical experiments in two and three dimensions confirm the predicted convergence rates and demonstrate the feasibility of the proposed criterion.

We show that the dispersionless version of the modified DKP hierarchy originally defined as the limit of relations for the tau-function of the Hirota-Miwa type has an equivalent reformulation as the Yang-Baxter equation for Baxter's $R$-matrix of Boltzmann weights for the 8-vertex model.

Let $F$ be a $k$-uniform hypergraph, abbreviated as $k$-graph. The codegree Turán density $\gamma(F)$ is the supremum over all $\gamma \in [0,1)$ such that, for arbitrarily large $n$, there exists an $n$-vertex $F$-free $k$-graph $H$ whose every $(k-1)$-subset of vertices lies in at least $\gamma n$ edges. Let $PG_m(q)$ be the projective geometry of dimension $m$ over finite field $\mathbb{F}_q$. In this paper, we prove that $\gamma(PG_m(q)) \ge \frac{1}{p}> 0$ for all $m$ and $q$, where $p$ is the smallest prime divisor of $q+1$. This resolves an open problem proposed by Keevash and Zhao (JCT-B, 2007). Moreover, we determine the exact codegree Turán density of $PG_4(q)$ when $q$ is an odd prime power.

Synchronization is ubiquitous across natural and synthetic systems, yet most prior studies focus on the inertia-free Kuramoto model and do so at the macroscopic level. In this study, we instead investigate the inertial Kuramoto model and analyze the kinetics of individual synchronized clusters that emerge in the underdamped dynamics, driven by the interactions among multiple synchronized clusters with different frequencies. Specifically, we explore two forms of intrinsic frequency distribution – unimodal Gaussian and multimodal uniform – and show that they give rise to qualitatively different synchronized clusters: a hierarchical organization for the Gaussian distribution and a homogeneous organization for the uniform distribution. This contrast leads to qualitatively different behaviors of the order parameter: for the Gaussian distribution, it increases smoothly with increasing coupling strength, while for the uniform distribution, it grows through a series of discrete jumps that trace out the size of the Devil's staircase (DS). By resolving the kinetics at the cluster level, we further find that the route to synchronization also depends on the distribution type: with a Gaussian distribution, a single dominant cluster forms and gradually entrains the remaining oscillators, whereas with a uniform distribution, synchronization proceeds via successive cluster mergers initiated from peripheral seeds associated with the high-frequency periphery. Taken together, these findings provide a new perspective on collective synchronization dynamics in inertial complex systems.

Real-world networks like the internet share patterns like a power law degree distribution and a high clustering coefficient. Many of these properties are captured by the generative model of hyperbolic random graphs (HRGs), which provides a theoretical framework for studying such networks. Motivated by the observation that several algorithms perform better on real-world networks than their worst-case guarantees suggest, we design and analyse distributed algorithms under the assumption that the input graph is an HRG. Indeed, prior work has shown that the classical symmetry-breaking problem of $\Delta+1$ colouring, where $\Delta$ is the maximum degree of the graph, can be solved in 2 rounds on HRGs [Maus and Ruff; SODA'26]. In stark contrast to this 2-round algorithm for $\Delta+1$ colouring, we prove that the related symmetry-breaking problems of maximal independent set (MIS) and maximal matching (MM) are substantially harder: we establish a lower bound of $\Omega\left(\frac{\log\log n}{\log\log\log n}\right)$ for MIS and MM on HRGs. Our lower bound techniques rely on new structural insights that may be of independent interest: we show that HRGs contain $d$-ary trees with large height and degree which enables us to adapt and lift prior impossibility results for distributed algorithms to the setting of HRGs. We also show that these lower bounds are polynomial tight: we design algorithms tailored to HRGs that solve MIS and MM in $\tilde{\mathcal{O}}(\log^{5/3}\log n)$ rounds with high probability in the LOCAL model, improving over the general worst-case lower bound of $\Omega\left(\min\left\{\log \Delta, \sqrt{\log n}\right\}\right)$ rounds [Khoury and Schild; FOCS'25].