Papers

We prove that the local limit of the minimum spanning arborescence in the complete graph (which is an oriented cousin of the minimum spanning tree in the complete graph) is the same as that of the uniform random tree, oriented towards infinity. The latter is known to be the critical Poisson Galton–Watson tree conditioned to survive. This is in sharp contrast with the local limit of the minimum spanning tree, which is known to be different from that of the uniform spanning tree (due to results of Addario–Berry \cite{AddarioBerry2013} and Addario–Berry, Griffiths and Kang \cite{PWIT_local_minimum}). Thus we demonstrate that introducing orientations changes the local geometry of the minimum spanning tree in a non-trivial manner.

The numerical simulation of non-Markovian open quantum systems based on the non-Markovian quantum state diffusion (NMQSD) equation is complicated by functional derivatives with respect to the stochastic process. A general numerical framework that directly treats these functional derivatives without relying on prescribed decompositions of the bath correlation function is still lacking. In this work, we derive an analytical solution of the linear NMQSD equation that reveals three elementary structures of the non-Markovian stochastic dynamics: stochastic propagation, functional-derivative insertion, and memory pairing. Based on this structure, we construct a general auxiliary-state framework for arbitrary bath correlation functions. The framework separates the numerical construction into time discretization, memory quadrature, and hierarchy truncation. We then construct first- and second-order schemes and provide diagrammatic transition rules for their explicit implementation. Numerical results verify the expected temporal accuracy and demonstrate the applicability of the proposed methods to different bath correlation functions and multi-level quantum systems.

Flow polytopes of directed acyclic graphs form a central class of lattice polytopes in algebraic, geometric, and enumerative combinatorics. Order polytopes are one of the best understood families of lattice polytopes; their Ehrhart theory, triangulations, volumes, and face structures are closely controlled by the combinatorics of the underlying posets. Mészáros–Morales–Striker proved that the flow polytope of an $st$-planar directed acyclic graph is unimodularly equivalent to an order polytope. In this paper, we prove a converse after contracting idle edges. More precisely, for a directed acyclic graph $G$ with a unique source and a unique sink, let $\widetilde G$ be the graph obtained from $G$ by successively contracting idle edges until none remain. We prove that $\mathcal{F}(G)$ is unimodularly equivalent to an order polytope if and only if $\widetilde G$ is $st$-planar. In addition, under a local three-good-neighbor condition, we prove that for a directed acyclic graph with a unique source, a unique sink, and no idle edges, the graph is $st$-planar if and only if it avoids an explicit list of forbidden butterfly minors.

We present numerical results for the problem of minimizing the first Dirichlet eigenvalue of the Laplacian among three-dimensional convex polyhedra with a prescribed number of facets. This problem can be viewed as a polyhedral version of the classical Rayleigh$-$Faber$-$Krahn inequality. Using a parameterization of polytopes by supporting hyperplanes, the problem is first reformulated as an equivalent finite-dimensional constrained minimization problem. A Lagrangian framework is employed for the numerical solution. Under the assumption that each vertex is incident to exactly $d$ facets, the sensitivity analysis of the Lagrangian combines finite-dimensional perturbations of the facets of $d$-dimensional polyhedra with infinite-dimensional shape derivatives of both the eigenvalue and the volume. Depending on the number of facets, the numerical results yield well-known polyhedra, such as Platonic solids, prisms, and the tetrakaidecahedron, or reveal nonregular optimal shapes exhibiting several symmetries.

In next-generation wireless networks, communication systems are expected to go beyond simple data transmission and simultaneously provide high data rates, efficiency, and security. This requirement has motivated the extensive adoption of machine learning methods to develop intelligent and real-time network management frameworks, enabling the system to continuously monitor and react to channel variations and user behavior while maintaining efficient information delivery. In this context, the integration of machine learning with beamforming enables adaptive and data-driven beam direction selection, improving both the efficiency and security of wireless links. In this work, a 3GPP-based system model is first implemented under a no-attacker scenario, and an exhaustive search is employed as a reference to identify the best beamforming configurations. The proposed framework is then evaluated in the presence of an attacker and under different network scalability conditions. We demonstrate that the reinforcement learning-based approaches, namely Q-learning and SARSA (State-Action-Reward-State-Action), consistently outperform random selection in terms of total channel capacity, attacker detection accuracy, and performance stability. Among the evaluated reinforcement learning methods, Q-learning achieves the best overall trade-off between detection accuracy and computational efficiency. Our results indicate that the proposed framework provides a stable, scalable, and effective solution for joint beamforming and security-aware decision-making in dynamic and adversarial wireless environments.

We reconsider the analysis of the Classical Painlevé Problem developed by F. Génot and B. Brogliato ([1] New results on Painlevé paradoxes. - European Journal of Mechanics-A/Solids, 18(4):653{677, 1999), focusing on the consistency of their results with Galilean invariance. We show that certain conclusions concerning the dynamical evolution of the mechanical system are not invariant under changes of Galilean observer and therefore cannot, in their present form, be interpreted as intrinsic properties of the system. In particular, we show that the classi?cation of motion states depends on the observer through the velocity{dependent characterization of the frictional constraint, and we trace the origin of this dependence to the Galilean velocity-addition theorem. We present and discuss possible reformulations of the model aimed at restoring observer-invariant descriptions of the problem.

We disprove Hoàng's conjecture that a minimally nonperfectly divisible graph cannot contain a bisimplicial vertex by constructing an explicit infinite family. For every integer $t\geq1$, the graph $G_t$ in this family has clique number three, contains a bisimplicial vertex of degree four, and satisfies \[ |V(G_t)|=93+30(t-1),\qquad |E(G_t)|=320+104(t-1). \] In particular, the members are pairwise nonisomorphic. The construction uses a fixed 15-vertex rooted graph and a variable auxiliary graph. Every copy of the rooted graph forces its identified root into the perfect part of every perfect division. Three induced odd holes in the auxiliary graph then force a triangle into the other part. A uniform assignment lemma and a rooted product lemma show symbolically that every proper induced subgraph of every $G_t$ is perfectly divisible. The finite properties of the fixed rooted graph are verified by exact exhaustive computation, with an independent implementation providing a cross-check. The construction also gives an infinite family of negative examples to a prescribed-vertex problem of Hu, Xu and Zhuang.

The orientable genus polynomial of a graph counts its cellular embeddings by genus. For finite simple $2$-connected cubic graphs it is a cycle-matroid invariant: $M(G)\cong M(H)$ implies $\Gamma_G=\Gamma_H$. The adjacency spectrum and the genus polynomial are incomparable: neither determines the other. We exhibit connected cubic graphs on $16$ vertices sharing the adjacency spectrum, spanning-tree count, girth, diameter, vertex and edge connectivity, automorphism-group order, and cycle counts through length $10$, yet with pairwise distinct genus polynomials. Splitting the expected face count at twice the girth explains the difference: short faces are spectral, long faces are not. We construct an explicit infinite family of connected cospectral cubic pairs $(G_t,H_t)$ on $14+2t$ vertices whose minimum genera differ. We also compute the genus polynomials of all $7,875,918$ connected cubic graphs through $22$ vertices and derive from short-cycle counts a deterministic lower bound on the minimum genus.

We study Kuznetsov-Shinder's categorical absorption for certain cone singularities, generalizing their results for nodal singularities. In particular, we give explicit descriptions of the endomorphism algebras of tilting objects for categorical absorptions of cones over certain Fano varieties admitting a geometric exceptional sequence, in the sense of Bridgeland and Stern. In the simplest “split case”, these algebras are truncations of certain Calabi-Yau completions in the sense of Keller. The split case occurs for anticanonical projective cones over many Fano varieties (like projective spaces, smooth quadrics, del Pezzo surfaces of degree greater than $4$, smooth del Pezzo threefolds of degree five, and finite products of these varieties). In general, the algebras are deformations of the split case. As a consequence, we obtain triangle equivalances between singularity categories of finite dimensional algebras and singularity categories of certain cone singularities, which also yields vanishing results in negative $\mathsf{K}$-theory. In a joint appendix with Yujiro Kawamata, we give an explicit description of tilting objects for weighted projective spaces $\mathbb{P}(1^d, m)$.

In this paper, we show a new lower bound for extreme values of quadratic Dirichlet $L$-functions with prime-related moduli, which generalizes the work of Darbar and Maiti in 2025, and sharpens a recent work of Gao in 2026.

Let \(N\) be a closed normal subgroup of a topological group \(G\), and let \(\widehat q:\rho G\to\rho(G/N)\) extend the quotient homomorphism. We prove that \(r_\rho(G/N)=\widehat q\bigl(r_\rho(G)\bigr)\) holds if and only if \(\widehat q\) is onto and \(\widehat q^{-1}(G/N)=G\), and we show that neither condition implies the other. Both conditions hold whenever \(N\) is Raikov complete and almost metrizable. Consequently, pseudocompactness of \(r_\rho(G)\) passes to \(r_\rho(G/N)\). In particular, the conclusion applies to closed locally compact normal subgroups.

In this short note, we prove that the category of necklaces is a test category. Hence presheaves on necklaces model homotopy types. The proof is analogous to that for the category of cubes with connections.

A hole is an induced cycle of length at least four, and an even hole is a hole of even length. A cap is obtained from a hole by adding a vertex adjacent to exactly two consecutive vertices of the hole. Chen, Xu, and Xu proved that every $\{\mathrm{cap},\mathrm{even\ hole}\}$-free graph $G$ satisfies $\chi(G)\leq \left\lceil\frac{5}{4}\omega(G)\right\rceil$, and improved this bound to $\chi(G)\leq \left\lceil\frac{7}{6}\omega(G)\right\rceil$ when $5$-holes are also excluded. They asked whether, for every integer $q\geq3$, every $\{\mathrm{cap},\mathrm{even\ hole}\}$-free graph $G$ with no odd hole of length at most $2q-1$ satisfies $$ \chi(G)\leq \left\lceil\frac{2q+1}{2q}\omega(G)\right\rceil. $$ We answer this question affirmatively and show that the bound is sharp for every $q\geq3$.

We show that the space of Bridgeland stability conditions Stab(Y) on a very general quintic threefold Y has infinitely many connected components. Using similar methods, we also prove that for a particular smooth projective (weak Fano) threefold X the braid group action on the set of full exceptional collections in Dscr^b(X) has infinitely many orbits, in particular, it is not transitive. This provides a counterexample to a conjecture of Bondal and Polishchuk from 1993. The conjecture was first disproved by Chang, Haiden, and Schroll, who constructed a family of partially wrapped Fukaya categories for which the transitivity fails. However, no counterexample of the form Dscr^b(X) for X a smooth projective variety was previously known. In addition, we show that Stab(X) also has infinitely many connected components. These are the first known examples of a smooth projective varieties whose spaces of Bridgeland stability conditions are disconnected.

At the beginning of this century, Langlands introduced a strategy known as Beyond Endoscopy to attack the principle of functoriality. Altuğ studied $\mathsf{GL}_2$ over $\mathbb Q$ in the unramified setting for the standard representation. We consider the case with ramification at $S=\{\infty,q_1,\dots,q_r\}$ with $2\in S$ and derive an asymptotic formula for the symmetric square representation adding some additional conditions on the test function so that the trace formula is simple. The limit is nonzero in general and we may detect the dihedral forms by using such limit form of the trace formula which is similar to Venkatesh's thesis. The proof involves a second Poisson summation, computation of the transformed Kloosterman sum and the corresponding series, and giving an asymptotic formula for the main term by residue analysis and using technical analysis to deal with the error term.

For integers $k\le n$, let $L_{k,n}$ be the set of words over $[n]$ of length at most $k$ in which no symbol is repeated. We present a nondeterministic finite automaton (NFA) of size $3.918^k n^{O(1)}$, improving on the $4^{k+o(k)}n^{O(1)}$ construction of Ben-Basat, Gabizon, and Zehavi. Our proof organizes several classical ingredients—product automata, hashing, and coefficient estimates—into a gadget-amplification framework: We take the product of many copies of a small local NFA gadget, whose language is a subset of $L_{r,c}$, and hash the $k$ input symbols to copies and local colors. The hash family guarantees that, for every repetition-free input, some hash sends at most $r$ symbols to each copy such that the resulting projection in every copy is accepted by the local gadget. Taking the nondeterministic union of the corresponding product NFAs yields a global NFA. Amplifying a $200$-state gadget for $L_{6,11}$ obtained from the small Witt design $S(4,5,11)$, this framework gives a $3.967^k n^{O(1)}$-size NFA. We then introduce the compose-and-compress technique, which deletes the expensive middle layers of these products and replaces paths across the deleted bands with sound one-symbol shortcut transitions. We apply it twice, once for enhancing the amplification framework and again for the local gadget, obtaining the stated result.

A commutative ring $R$ is called a Gröbner ring if, for every $n\geq 1$, the leading term ideal of every finitely generated ideal of $R[X_1,\ldots,X_n]$ is finitely generated with respect to the lexicographic order $X_1\succ\cdots\succ X_n$. We prove that this property is equivalent to the condition that, for every $n\geq 1$, every rational monomial order $\prec$ on $R[X_1,\ldots,X_n]$, and every finitely generated ideal $I\subseteq R[X_1,\ldots,X_n]$, the leading term ideal $\operatorname{LT}_{\prec}(I)$ is finitely generated. The construction uses a tagged monomial embedding, a compatible group grading, and dehomogenization. As an application, we apply this reduction to valuation rings and determine when the finite generation property holds for every rational monomial order. In particular, for valuation domains, this gives a proof of the rational monomial order version of the Gröbner ring conjecture.

Fluid antenna arrays (FAAs), support multiuser downlink transmission by activating a subset of reconfigurable ports. The activation mask jointly determines the effective channel and the sparse radiating aperture, which requires a balance among sum rate, sidelobe suppression, hardware constraints, and online complexity. Channel driven selection can cluster active ports and increase sidelobes, whereas sidelobe oriented synthesis is typically channel independent and can sacrifice sum rate. This paper proposes learned blockwise port activation (L-BPA), for real time sidelobe aware FAA downlink beamforming. L-BPA activates a fixed number of ports in each aperture block, which supports grouped switching hardware and limits port clustering. A lightweight convolutional network scores ports using multiuser channel features, port coordinates, and user power statistics. Training combines blockwise straight through masks with a differentiable peak sidelobe level (PSLL), surrogate. During inference, learned scores are combined with multiscale geometric repulsion, followed by regularized zero forcing precoding over the reduced effective channel. L-BPA reduces the average PSLL by 3.26 dB relative to uniform sparse activation while achieving a slightly higher sum rate. It also reduces the PSLL by 8.13 dB and 10.10 dB relative to greedy and gain based selection, respectively, without iterative online search.

Semantic integrated sensing and communication (ISAC) is envisioned as a promising paradigm for efficient and intelligent connectivity in future wireless networks. However, the open wireless channel exposes the dual-functional waveform to detection, which challenges the joint guarantee of covertness, sensing fidelity, and semantic accuracy. To address the challenge, we propose CoSMIC, a novel covertness-oriented semantic ISAC framework, where the sensing output is embedded into a dual-functional ISAC waveform through semantic modulation. Specifically, a semantic rotation coding scheme is established to map semantic latents onto the pairwise rotation and scaling of Gaussian reference sequences, which satisfies a derived closed-form covertness constraint by a differentiable budget projection. Moreover, the radar performance is analyzed to confirm an invariant matched-filter mainlobe response and a bounded output signal-to-interference-plus-noise ratio (SINR) under the semantic embedding. Subsequently, a reliability-guided rectified flow (RFlow) refiner is designed to effectively reconstruct high-fidelity semantic representations from coarse observations. Simulation results demonstrate that CoSMIC improves the semantic reconstruction quality by 18% over diffusion-based baseline schemes with substantially reduced inference latency under strict covertness constraints, which validates the applicability to practical ISAC scenarios. The source code and video demonstrations are available at https://github.com/LanceAnlan/CoSMIC-covertness-oriented-semantic-ISAC-framework.

We study the classical topological dynamical notions of shadowing and topological stability from a viewpoint of paired dynamical system $(f,g)$, where $f$ and $g$ are uniform equivalences on a metric space $X$. We observe that if $g$ is equicontinuous and commutes with $f$, then the study of shadowing for $(f,g)$ is reduced to the study of the classical shadowing for $g^{-1}f$. The fact that these assumptions are sufficient is justified through examples. Finally, we prove that if $f$ is an expansive homeomorphism on a relatively compact metric space, then any pair $(f,g)$ with shadowing is topologically stable.

Let $(M^4,g)$ be a complete four-dimensional Riemannian manifold. First, if the $Q$-curvature $Q_g\geq 6k^2$ and scalar curvature $R_g\geq -12k$ for some positive constant $k$, then $(M^4,g)$ is either Einstein with $Ric_g=-3kg$ or compact with $R_g\ge 12k$. As a corollary, the fundamental group $\pi_1(M^4)$ satisfies $|\pi_1(M^4)|\leq 16\pi^2/(\int_{M^4}Q_g d\mu_g),$ under the additional assumption $R_g>-12k$. Second, if the scalar curvature $R_g>0$ and $Q_g\geq \theta R_g$ for a positive constant $\theta$, then $M^4$ is compact and the diameter of $(M^4,g)$ is at most $4\pi/\sqrt{15\theta}$.

We study the asymptotic behavior of the convex core volume for a sequence of quasi-Fuchsian manifolds $Q(\psi^{-n}X, \psi^n X)$ associated with a pseudo-Anosov mapping class $\psi$ on a once-punctured torus. We prove that the volume of the convex core differs from $2n$ times the volume of the mapping torus of $\psi$ by at most a uniformly bounded constant.

We show that rim surgery on a smooth, oriented, properly embedded surface in $B^4$ does not change the map on Khovanov homology induced by the surface, answering a question raised by Hayden and Sundber. More generally, the same insensitivity holds for a class of local surgery operations that we call local annulus replacements. Combined with the work of Hayden–Sundberg, this result yields exotic pairs of surfaces in $B^4$ that cannot be related by any sequence of local annulus replacements. We give two proofs, both making essential use of Khovanov skein lasagna modules.

This paper deals with parabolic equations associated with double phase functionals. It is known that double phase functionals may exhibit the Lavrentiev phenomenon, which indicates an existence of a singular minimizer. Our aim is to investigate the process of the associated double phase flow evolving toward the singular minimizer. For this purpose, we study whether solutions can be approximated by smooth functions. We first observe a phenomenon that we call finite-time loss of smooth approximability; more precisely, we prove that the flow eventually ceases to be smoothly approximable. We also establish quantitative estimates on the time of loss. On the other hand, we investigate a phenomenon that we call short-time persistence of smooth approximability; we prove that the smooth approximability persists for a short time provided that the initial datum is regular and that the functional is nondegenerate with respect to the gradient variable. The results concerning finite-time loss of smooth approximability are derived from the evolution variational inequality, whereas the short-time persistence result is obtained by applying analytic semigroup theory. The novelty of this paper lies in studying the dynamical aspect of the Lavrentiev phenomenon, which is usually regarded as a stationary phenomenon.

This paper studies finite-time blow-up of pseudospherical metrics induced by solutions of the Cauchy problem for the Degasperis–Procesi equation. For initial momentum profiles satisfying suitable left–right sign conditions, we use the method of characteristics, the momentum-transport formula, the Green's-function representation, and Riccati-type differential inequalities to analyze the metric along a distinguished characteristic. We prove that the associated coframe remains non-degenerate before the critical time, so that the induced pseudospherical metric is well defined in the precritical region. Moreover, as wave breaking is approached, the metric component \(g_{22}\) diverges to \(+\infty\); when \(\mu\neq0\), the mixed component \(g_{12}\) also blows up in absolute value. Thus, finite-time wave breaking for the Degasperis–Procesi equation is shown to induce blow-up of certain components of the corresponding pseudospherical metric.

We study the behaviour of the Brauer–Manin obstruction to the existence of rational points under finite field extensions. For $(p, mp)$-normic bundles over number fields, we prove that the Brauer–Manin obstruction vanishes after base change to finite extensions whose degrees satisfy suitable $p$-divisibility conditions depending on $m$. We further show that, for $(p,2p)$-normic bundles with $p=2$ or $3$, it is enough to assume that the extension degree is divisible by $p$. We also prove that the divisibility hypothesis is, in general optimal, by constructing a conic bundle for which the Brauer–Manin obstruction persists over a quadratic extension.

We study coefficient problems for the class $\mathcal{P}^{*}$ of normalized analytic functions $f$ in the unit disk satisfying the subordination condition \[ f'(z) \prec e^z(1+\sin z), \qquad z\in\mathbb{D}. \] For functions in this class, we determine sharp bounds for the logarithmic coefficients $\gamma_n$ for $n=1,2,3,4$ and for the inverse logarithmic coefficients $\Gamma_n$ for $n=1,2,3$. In particular, we prove \[ |\gamma_n| \le \frac{1}{n+1} \quad (n=1,2,3,4), \qquad |\Gamma_1|,|\Gamma_2| \le \frac12,\quad |\Gamma_3|\le \frac{35}{48}, \] with equality cases explicitly identified. Moreover, we establish sharp estimates for the moduli of the differences $|\gamma_2|-|\gamma_1|$ and $|\Gamma_2|-|\Gamma_1|$, yielding \[ -\frac12 \le |\gamma_2|-|\gamma_1| \le \frac13, \qquad -\frac{1}{\sqrt{10}} \le |\Gamma_2|-|\Gamma_1| \le \frac13. \] Turning to determinant problems, we obtain sharp upper bounds for the second-order Hankel determinants associated with both the logarithmic and inverse logarithmic coefficients: \[ \bigl|H_{2,1}(F_f/2)\bigr| \le \frac19,\qquad \bigl|H_{2,1}(F_{f^{-1}}/2)\bigr| \le \frac{11}{96}. \] Finally, we derive the sharp two-sided estimate \[ -\frac{9}{35} \le T_{3,1}(f) \le 1 \] for the third-order Hermitian–Toeplitz determinant. All bounds are sharp, and the extremal functions are given explicitly. Our results extend and refine several known coefficient estimates for related subclasses of starlike and convex functions.

Why do ChatGPT-like AIs, despite major architectural and training differences, unexpectedly tip to undesirable content (e.g. harmful, misleading, repetitive) even under deterministic greedy decoding? We show that a broad class of such tippings is caused by the many-body interactions between tokens (spins) as they cross the finite-layer system. Tipping emerges as a dynamical first passage process between competing output basins. Attention disorder controls the transport toward, away from, or along the basins' boundary. A few-basin reduction yields a closed finite-layer threshold, whose coarse-grained predictions show good agreement across ChatGPT-like families. These results suggest that a broad class of AI failures represents 'foreseeable engineering risk' rather than inherently unpredictable behavior, with important implications for legal and societal assessments of AI harm.

Let n be a natural number. It is proved that if the sum of divisors of n has sufficiently large relative to n, then n is some distinct sum of proper divisors of n.

This note considers a constrained-degree percolation model in a random environment on the square lattice, where each vertex is assigned a random degree constraint, and edges attempt to open at random times. We prove that whenever an infinite open cluster exists, it is almost surely unique, implying the positivity of the connectivity function in the supercritical regime.