Papers

In this paper, we explore the idea that the graded Grothendieck group $K_0^{gr}$, or equivalently its positive cone, the talented monoid, can detect the structural type of higher-rank graph algebras (i.e., higher-rank graph $C^*$-algebras and Kumjian–Pask algebras). We show that the talented monoid captures some of the essential geometric information of a higher-rank graph, including the existence of cycles with and without entrances. In turn, we show that the graded $K$-theory can effectively distinguish the class of locally finite Kumjian–Pask algebras, and also the class of crossed product Kumjian–Pask algebras. We also derive talented monoid criteria for higher-rank graph algebras to be purely infinite simple, and not to be $AF$ or ultramatricial.

We develop a functional-analytic framework for smooth directional point descriptors in GPS-free map-based localization. Given a query point $\mathbf{p}$ in a map $\mathcal{M} \subset \mathbb{R}^d$, the descriptor integrates an environment signal against a partition-of-unity weight family built from a softmax kernel, yielding a $\mathcal{C}^\infty$ alternative to hard angular binning. Our main contributions are: (i) a totality theorem showing that the associated linear functionals form a total family in $L^2(\mathbb{S}^{d-1})$, establishing asymptotic injectivity of the descriptor map; and (ii) a descriptor-induced seminorm $|f|_{\mathcal{D},n} = \|P_n f\|_{L^2}$, identified via the Gram matrix of the weights, which satisfies a Parseval-type identity $|f|_{\mathcal{D},n} \to \|f\|_{L^2(\mathbb{S}^{d-1})}$ as $n \to \infty$. Complementary results include Fréchet differentiability, lower semicontinuity under occlusion, and explicit Lipschitz stability bounds with constants depending on the kernel and temperature. These properties underpin a localization theory in which the descriptor grid enables nearest-neighbor position recovery with a certifiable static error bound, while robot motion generates an observability Gramian whose smallest eigenvalue controls a dynamic error bound and generically resolves symmetry-induced ambiguities that persist under single-observation matching.

What a finite learning device has recorded and what will hold value for it on future tasks are not the same quantity. We develop a typed accounting for finite-state learning devices that separates four components: a training-side fit functional $\Phi_{\mathrm{fit}}$, the record-correlation stock $J_{D}=I(M;D)$, an update-side search ledger $\sigma_{M}$, and an operational capital value $V(M;T,b)$. This value is the work gap between an informed protocol class and a blind class obtained by deleting the memory-read port and re-optimizing from scratch. (I) Separation: for every $n$, there is a device family on which record correlation and world correlation grow by $n\ln 2$ while the capital gain is exactly zero. In the $\mathrm{flat}^{*}$ regime, data-free updates never increase $V$. (II) Capitalization ledger: an exact $\mathrm{flat}^{*}$ extraction identity and a universal ledger identity give, for (F5$'$)-stable $M$-local updates under a no-discarded-record-correlation condition (f), the bound $\eta_{\mathrm{cap}}\le 1$ for the capitalization efficiency $\eta_{\mathrm{cap}}=\Delta V/(k T\,\sigma_{M})$, together with necessary and sufficient conditions for equality. (III) Value retention: for the retention gap $L_{\mathrm{gen}}$ and retention ratio $\rho_{\mathrm{gen}}$ (the former carries no sign constraint; the latter is defined for positive training-side value and is not confined to $[0,1]$) we give a two-layer alignment domain: an exact exchange rate between value and the side-information-adjusted record fit $I(M';D\mid Y)$ without any record-side-information independence assumption, and a raw record-stock exchange rate under a joint side-information neutrality condition $(M,D)\perp Y$, whose boundary is marked by an explicit one-time-pad witness. These are statements about finite-device value retention under task-distribution shift, not a theory of statistical generalization.

In this paper, we develop the thermodynamic formalism for intermittent full-branch interval maps with finitely many neutral fixed points and for a broad class of possibly discontinuous potentials. Under a hyperbolicity assumption on the potential, we prove the existence of a unique equilibrium state with positive entropy and exponential decay of correlations. Furthermore, we characterize the phase transitions for the one-parameter family $\beta\varphi$, with $\beta>0$, and prove that, at the critical parameter, multiple ergodic equilibrium states may coexist, at most one of which has positive entropy. Our approach relies on a detailed analysis of conformal measures.

In this short note, we address problems concerning the characterization of simple and related groups by various arithmetic invariants. We propose a uniform approach to such questions and discuss both known results and open problems in this area.

Given a family of graphs $\mathcal{F}$, we define a graph $G$ to be fully $\mathcal{F}$-colourable if $G$ admits a full homomorphism to some $F$ in $\mathcal{F}$. We approach the problem of determining when a graph is fully $\mathcal{F}$-colourable in terms of minimal forbidden induced subgraphs. We provide general results which allow to obtain the exact families of forbidden induced subgraphs for full $\mathcal{F}$-colouring when $\mathcal{F}$ is among some well-known families, such as threshold, trivially perfect, split, chordal, interval and strongly chordal graphs, as well as forests. Traditionally, these questions have been studied for a single graph $H$, not a family. Motivated by our results on the family of forests, we contribute to this research by focusing on the case of a single centipede.

Global warming, driven by anthropogenic carbon emissions with transboundary pollution characteristics and irreversible damage, poses an existential threat to human society. This paper develops a novel two-level Stackelberg game with mean field interaction of controls and common noise which integrates hierarchical decision-making under a state-reflected emission dynamics. A central regulator (leader) adjusts product prices to guide $n$ heterogeneous competing regions (followers) while enforcing a hard emission cap via a reflection mechanism that models emergency reductions through a local time process. We establish the existence of an approximate Stackelberg equilibrium and perform sensitivity analysis via Monte Carlo simulations.

A similarity pair is the dynamical system in $\mathbb{C}$ generated by two maps $f:z \to sz-1$ and $g:z \to sz+1$ for $|s|<1$. Associated to the dynamical system is an attractor $\Lambda$. The Barnsley–Harrington Mandelbrot set $\mathcal{M}$ is the set of $s\in \mathbb{D}$ for which $\Lambda$ is connected. Let $K$ denote the filled set of a connected $\Lambda$. For $s\in \partial \mathcal{M}$ we show that the (partially defined) action of the semigroup on $\partial K$ is topologically conjugate to a (discontinuous) piecewise linear action of constant slope. Conditional on a conjecture (satisfied for `most' $s\in \partial \mathcal{M}$) we give a necessary and sufficient condition in terms of the dynamics on $\partial K$ for $K$ to contain cut points, and we describe the set of all such cut points in terms of infinite walks in a directed graph $\textrm{IG}$ obtained by an explicit recursive algorithm. The structure of the `dynamical cut point set' for a 2-dimensional family of piecewise linear actions (containing those coming from $s\in \partial \mathcal{M}$) recovers and generalizes the Douady–Hubbard–Thurston quadratic minor lamination for the abstract Mandelbrot set.

This paper is devoted to the introduction and analysis of a penalty-type method for the numerical treatment of a class of bilevel optimization problems arising from inverse optimal control. The algorithm is designed to compute stationary points of the associated relaxed value function reformulation. This is achieved by determining a sequence of stationary points associated with a sequence of surrogate problems where the relaxed value function constraint is penalized, where the updates of upper- and lower-level decision variables are decoupled, and where the penalty parameter is enlarged only in those iterations which do not come along with a sufficient improvement of some feasibility measure. The resulting method does not comprise any linesearch, the lower-level problem has to be evaluated just once per iteration, and the penalty parameter does not need to be driven to infinity. Nevertheless, subsequential convergence results are obtained under reasonable assumptions. Numerical experiments, where the relaxation parameter is also driven to zero, visualize effectiveness of the approach.

In-space servicing has been receiving great attention to extend the operation of spacecraft with defective components. This requires rendezvous and proximity operations for a chaser to provide service to a target. This work constructs a fuzzy inference system-based controller for the chaser to reach the cooperative target on a circular orbit in the final approach phase while minimizing the energy consumption of the chaser. The offline training process performed by a genetic algorithm deals with multiple initial relative positions of the chaser, and the trained controller is validated using a testing environment with disturbances, which differs from the training scenarios.

Let $f(t)\in\mathbb{Z}[t]$ be a nonconstant polynomial with nonzero discriminant and let $k\ge2$ be an integer. Inspired by the work of Alon and Bourgain, for sufficiently large prime $p\equiv1\pmod{k}$, we study cyclic orderings of subsets $A\subseteq \mathbb{F}_p$ for which $ f(a_i+a_{i+1})$ is a nonzero $k$-th power for every consecutive pair. By combining mixed character-sum estimates, Fourier analysis on $\mathbb{F}_p$, and spectral graph methods, we establish a threshold $c(p,k,f)$ such that every subset $A$ with $\#A\ge c(p,k,f)$ admits such a cyclic ordering. We also give lower and upper bounds for the optimal threshold.

Laplacian-regularized minimization is fundamental in signal processing and machine learning, but explicit construction of the dense graph Laplacian pseudoinverse can be computationally expensive. To address this issue, we consider the setting where the graph Laplacian is given and propose a Difference-of-Convex Regularizer (DCR) graph learning framework that learns a reusable approximation of the pseudoinverse action through regularized Maximum Likelihood Estimation (MLE) and shrinkage-regularized Convex–Concave Procedure (CCCP) iterations. For Laplacian-Regularized Nonnegative Least Squares (LR-NNLS), dual and Karush-Kuhn-Tucker (KKT) analysis motivates separating graph-dependent pseudoinverse learning from instance-specific optimization. The learned pseudoinverse is embedded into a differentiable primal optimization procedure as a graph-aware preconditioner for nonnegative reconstruction and can be reused across LR-NNLS instances sharing the same graph. We establish stability and fixed-point existence guarantees for the proposed pseudoinverse-learning iteration. Experiments across different graph topologies and scales demonstrate near-reference accuracy, competitive time-to-moderate-accuracy behavior, and effective cross-instance reuse across independently generated problem instances sharing the same graph.

In this paper, we study the observability and controllability of a class of degenerate hyperbolic equations with a control region intersecting the degenerate set. Unlike the existing results that mainly deal with control regions separated from the degeneracy, we consider the case where the control region reaches the degenerate part. To handle the difficulty caused by the degeneracy, we introduce a shape-design-based approximation method based on shape design by approximating the degenerate equation with a family of uniformly hyperbolic equations. The proof relies on the spectral approximation of the associated operators, precise estimates for weak solutions, and the multiplier method. We first establish observability inequalities for the approximating equations with constants independent of the approximation parameter. Then, by passing to the degenerate limit, we obtain the observability inequality for the original degenerate equation.

Domination of gd-sets, the relation recording which function family approximates which, is closed under box convergence. More generally, approximate 1-Lipschitz maps on asymptotically full-measure subsets, whose restricted measures converge to the target measure after pushforward, place the target representation in every subsequential weak limit of the source pyramids. For weakly convergent pyramids, bounded joint distributions of finite ordered tuples of observables recover both observable diameter, after rightward perturbation of its mass parameter, and the largest common forward gap among several positive-mass sets, after common leftward perturbation of their mass parameters. The lower- and upper-limit formulas agree as the perturbations vanish. Without closure assumptions on a gd-set's function family, we compare observable diameter, the nonnegative part of forward separation, and upper and lower median-tail masses. All observables become uniformly close in measure to suitable constants exactly when both median-tail masses vanish at every positive radius.

This paper studies the exterior Dirichlet problem for Hessian quotient equations with nonconstant right-hand sides. We prove the existence of classical admissible solutions with prescribed asymptotic Hessians and establish convergence of the Hessian at infinity. The main difficulties are obtaining second-order estimates on expanding annuli that are uniform in the outer radius and deriving Hessian convergence under an integral tail condition with no prescribed decay rate. These are resolved through a radius-independent boundary-to-interior estimate and a blow-down argument. We also allow nonradial perturbations of the source. Under stronger pointwise assumptions, we obtain higher-order asymptotic expansions and solutions for every sufficiently large prescribed asymptotic constant.

We develop a stable convergence theorem for partial sum processes on sample-size dependent stochastic bases. The result allows multidimensional semimartingale limits that have conditionally independent increments and both a continuous and discontinuous martingale part. Motivated by infill asymptotics, it complements classical Gaussian stable limit theorems and supports applications to likelihood based statistical inference.

In the present article, we consider two families of integer sequences, the $(k,r)$-Lorenz-Fibonacci sequences of the first and second kind, whose characteristic polynomial is $$p_{k,r}(x)=x^k-x^{k-1}-\cdots- x^r+x^{r-1}+\cdots +x+1,$$ where $k\geq 2r+1$, and $k\geq 3$. These families include the well-known $k$-bonacci sequence, when $r=0$. These sequences arise naturally in the study of the dynamics of the Lorenz attractor. We introduce two families of substitutions (in an alphabet of $k$ symbols) so that they are associated with each of the families of integer sequences and share the same polynomial. We study the main combinatorial properties of these substitutions.

We develop a new uniform drift condition and local minorization that implies a stronger weighted form of uniform ergodicity for Markov chains we call hyper-V uniform ergodicity. The convergence guarantees geometric decay of the bias towards the invariant measure independently of the initialization for all functions controlled by a dominating function V. A key advantage of the approach is that it bypasses the need to establish a global minorization condition, which is often substantially more difficult to verify in practice, while yielding stronger convergence guarantees than global minorization. Optimal convergence bounds in a minimax sense of the framework are established. The utility of the framework is demonstrated through applications to the P'olya-Gamma and Kolmogorov-Gamma Gibbs samplers. We also show qualitative hyper-V uniform ergodicity convergence for two-variable Gibbs samplers can be inferred by the form of the invariant measure, bypassing convergence analysis entirely.

We study positive solutions of the critical fractional Hartree equation with a non-removable isolated singularity at the origin. This equation is doubly nonlocal, involving both the fractional Laplacian and a Riesz convolution potential. We first prove that every positive singular solution is radially symmetric about the origin, by combining the Caffarelli–Silvestre extension with the method of moving spheres. We then establish the existence of Delaunay-type periodic singular solutions. After the Emden–Fowler transformation, the Hartree convolution survives as a genuinely nonlocal integral term, so that the resulting periodic equation cannot be reduced to an ordinary differential equation. We construct nonconstant periodic solutions for all sufficiently large periods by minimizing a Rayleigh-type quotient in a periodic fractional Sobolev space.

The inverse spectral problem for the radial Schrödinger operators on the finite interval is investigated. The potential is recovered from the given eigenvalues plus its information on a smaller interval. The method is to discuss the connections between the Paley-Wiener spaces and the potentials, from which we completely determine the potential on the whole interval from the potential on a smaller interval and a set of eigenvalues in terms of the complete exponential systems.

We give a convex-geometric formula for the multiplicity sequence of a monomial ideal in terms of mixed volumes of polytopes constructed from its Newton polyhedron. We also construct a counterexample to a conjecture of Achilles and Manaresi proposing a different volume formula for the multiplicity sequence. Finally, we derive a mixed-volume formula for the mixed multiplicities of arbitrary monomial ideals.

We develop the first rigorous topological framework for an auxiliary boundary-integral model motivated by physical scaling identities for wave scattering from self-similar or fractal media. The model is periodic in logarithmic scale, and a multiplicative Bloch-Floquet-Zak transform fiberizes its interscale coupling. For sufficiently small, well-separated components, equilibrium densities define a computable finite-dimensional projected matrix, while Riesz projections select the corresponding invariant spectral subspaces of the full boundary symbol. Under suitable spectral-isolation and point-gap conditions, and after recentering the two families at their respective same-scale reference values, we prove that the determinant winding of the exact Riesz-reduced family agrees with that of the projected matrix family. For regular-simplex configurations, we derive explicit nonzero winding formulas and track the resulting local winding data across finite prefractal levels and along a geometric sequence of wavenumbers. We also formulate conditional winding data for multiple dilation centers and show that the Zak phase of the chiral Hermitianization equals $\pi$ times the point-gap winding modulo $2\pi$. Thus, the scale-periodic boundary model admits a rigorous topological reduction.

We construct rank-zero irregular vertex operators for the Neveu–Schwarz algebra as linear maps between irregular Verma modules of the same rank satisfying the usual superconformal commutation relations and an irregular asymptotic condition. Under the nondegeneracy assumption $\Lambda_{2p}\neq0$, we prove their existence and uniqueness. We then extend the decomposition theorem for the Neveu–Schwarz algebra to the irregular setting: the tensor product of the free-fermion Fock module with a rank-$p$ Neveu–Schwarz irregular Verma module decomposes into an infinite direct sum of tensor products of two Virasoro irregular Verma modules. We also prove a compatible decomposition of the irregular vertex operators into tensor products of Virasoro irregular vertex operators. Using suitable pairings and mode insertions, we derive bilinear differential equations for weighted sums of products of Virasoro irregular conformal blocks of types $(0,0,1)$ and $(0,2)$. After explicit parameter identifications and a gauge transformation in the Painlevé V case, the resulting bilinear differential operators agree with those appearing in the quantum Painlevé V and IV tau-function equations.

Wood~ recently showed that every graph $G$ of pathwidth $h$ and $\Delta(G)\ge1$ admits a $T$-partition of width at most $4(h+1)^2\Delta(G)$ for some tree $T$ with $pw(T)\leq2h+1$. In this paper, we establish an analogous result for tree-depth, which is a stronger parameter than pathwidth. We prove that every connected graph with tree-depth $h$ admits a $T$-partition of width at most $\mathrm{max}\{1, (4h-10)\Delta(G)+1\}$ for some tree $T$ with $\operatorname{rad}(T)\leq h-1$.

We prove that the archimedean Asai $L$-factor attached to an irreducible generic representation of $\operatorname{GL}_n(\mathbb{C})$ can be expressed as a finite sum of Flicker local zeta integrals. For an archimedean local field $F$, we also prove that the exterior-square $L$-factor attached to an irreducible generic representation of $\operatorname{GL}_m(F)$ can be expressed as a finite sum of Jacquet–Shalika local zeta integrals.

Muon, a more recently developed optimizer, is useful for matrix-wise models in AI areas. Although many works have studied Muon and its variants, these methods are still not particularly well-suited for hierarchical structured problems. To fill this gap, we propose an effective federated compositional Muon (FedCoMuon) optimizer to solve distributed matrix-wise compositional optimization problems. Specifically, our FedCoMuon optimizer builds on compositional gradient tracking and orthogonalized momentum. Moreover, we propose a variance reduced variant of FedCoMuon (FedCoMuon-VR) based on a momentum-based variance reduced technique. In theory, we analyze the convergence properties of our algorithms under the non-i.i.d. and non-convex settings. In particular, we prove that our FedCoMuon-VR obtains a lower sample complexity of $O(\epsilon^{-3})$ for finding an $\epsilon$-stationary solution than the existing FedMuon algorithms. Extensive numerical experiments on robust federated learning and task-distributed risk-sensitive meta learning show that our proposed methods are competitive with existing compositional baselines and achieve the best reported accuracy in several settings.

Enrique González-Jiménez and Álvaro Lozano-Robledo proved that there exists a uniform bound on the levels $n$ for which the $n$-division field $\mathbb{Q}\left(E\left[n\right]\right) / \mathbb{Q}$ of an elliptic curve $E/\mathbb{Q}$ defined over $\mathbb{Q}$ is abelian. Assuming GRH (Generalized Riemann Hypothesis), Allen and Genao partially generalized this result by restricting to prime levels while allowing arbitrary number fields without rationally defined CM. In this paper, we remove the GRH assumption and prove that the property established by Allen and Genao is in fact equivalent to the absence of rationally defined CM.

We study whether the topology underlying a Calabi-Yau manifold can support natural geometric positivity or negativity structures. In even complex dimension $n \geq 4$ (assuming $b_2=1$ when $n \geq 6$), we strengthen a theorem of Oguiso-Peternell by proving that a Calabi-Yau manifold is not homeomorphic to a weak Fano $n$-fold. The same obstruction applies to Kähler manifolds with quasi-positive holomorphic sectional curvature. A transformation-group analogue excludes, in particular, symplectic manifolds admitting Hamiltonian circle actions with isolated fixed points. On the negative side, we show that the fundamental group of a Calabi-Yau manifold is not isomorphic to that of a Kähler hyperbolic manifold. Taken together, these results exhibit a common topological rigidity separating Calabi-Yau manifolds from several fundamental classes governed by geometric positivity or negativity.

Multi-objective bilevel optimization has wide applications in the AI area such as automated learning and multi-task meta-learning. Although recently some works have been begun to study the multi-objective bilevel optimization, the proposed methods rely on the (strongly) convex lower level problems. In fact, these multi-objective bilevel learning problems are generally nonconvex, and particularly their lower level problems are nonconvex. To fill this gap, we propose a class of Multi-Objective Moreau Envelope based Hessian-free Algorithms (MOMEHA) for the multi-objective bilevel learning problems with nonconvex lower level. Specifically, our method uses the Moreau envelope to relax the original problem into a multi-objective single-level optimization with an envelope constraint. In particular, our method retains computational advantages of being single-loop and Hessian-free in the multi-objective setting by incorporating a smooth weighted Tchebycheff scalarization. Furthermore, we propose a momentum-based variant of MOMEHA (i.e., MB-MOMEHA) method for the stochastic multi-objective bilevel learning problems. In theory, we provide the convergence properties of our algorithms under both deterministic and stochastic setting. Some experiments on few-shot meta-learning and neural architecture search demonstrate that our methods outperform the existing approaches in Pareto front, validating its effectiveness and robustness.

The necessary conditions for formulating optimal feedback control algorithms have been known for many years. Free parameters exist in the performance index in the form of state and control penalty and terminal state penalty matrices for tuning the performance of the optimally controlled system. The selection process is typically experimental and iterative. To generate the weight matrices for optimal feedback control algorithms, an optimization process is proposed. Typically, the optimization process for the weight matrices requires several numerical integration processes that are computationally expensive; this work overcomes the classical high computational cost by exploiting closed-form solutions for the time-varying Riccati matrix, state trajectories, state transition matrix, and the optimal performance index. Closed-form algebraic equations are used to generate all partial derivative calculations, and no numerical integration is required. The closed-form partial derivatives are used to generate analytic gradients for the optimization steps. The optimization strategy seeks to minimize the terminal state values for the feedback control problem. A numerical example is presented to demonstrate the effectiveness of the proposed optimization algorithm. The resulting computational procedures are expected to be broadly useful for control theory applications in science and engineering.