Papers

The class of $q$-Bass martingales provides a natural answer to a central question in martingale optimal transport: how to construct martingales with prescribed initial and terminal marginals whose transition kernel remains as close as possible to a given reference measure $q$. We prove the existence of $q$-Bass martingales when the initial marginal is supported on finitely many atoms, and establish uniqueness, up to an additive translation constant, of the associated Bass measure. Our approach is geometric and relies on the analysis of a suitable parametrization of convex polygonal chains.

In this paper, we define and study partial orders on a Rickart *-ring obtained by imposing an additional condition on the star and the one-sided star partial orders. For each such order, we show that the down-set of any element is order-isomorphic to a suitable subset of self-adjoint idempotent elements. As an application, we characterize the elements which are below a given element with respect to each of the new orders. We further prove that the down-set of any element is a lattice whenever the ring is regular. We analyze the existence of supremum and infimum of pairs of elements in a regular Rickart *-ring and provide characterizations of these operations whenever they exist. Finally, we extend the latter results for a nonempty subset of elements for regular Baer *-rings.

This work investigates the existence and classification of almost complex structures and Kähler metrics on principal bundles, with particular emphasis on the homogeneous setting. Using Wang's theory of invariant connections, we give a direct geometric proof of the classification of invariant integrable complex structures established by Biswas and Upmeier. This approach avoids the algebraic machinery of Jordan triple systems and extends the classification from Hermitian symmetric spaces to general symmetric spaces admitting invariant complex structures. We also present a computationally direct proof of Johnson's Kähler criterion using the Levi-Civita connection. Finally, we apply these classification results to the reduced frame bundles of the upper half-plane and complex projective spaces, showing that the invariant integrable complex structures in these cases are unique.

Urban transit electrification requires balancing energy efficiency, schedule reliability, and ride comfort for electric buses (EBs), particularly when interacting with transit-priority adaptive signals in congested networks. This paper proposes PriEco-DRL, a joint optimization framework that integrates EB eco-driving with transit-priority adaptive signal control using deep reinforcement learning (DRL). The signal layer employs a priority-weighted max-pressure (Priority-MP) controller to allocate green time based on occupancy-aware pressures, while the vehicle layer adapts longitudinal control based on uncertain and dynamically evolving local signal cues. A structured reward combines guidance and event-based reinforcement to align EB arrivals with green opportunities while considering energy, time, comfort, and safety. The framework uses centralized training and decentralized execution (CTDE) with parameter sharing, allowing a single DRL agent to learn from multiple buses and routes using local observations. Experiments on a real-world corridor show that PriEco-DRL reduces EB energy consumption while maintaining network efficiency and transit priority compared with fixed-time, actuated, and rule-based signal-vehicle coordination baselines. Energy- and trajectory-based analyses reveal that the improvements stem from fewer unscheduled stop-start events and smoother speed regulation under adaptive signals. The results highlight a tunable energy-time trade-off, allowing flexible operational choices through reward weighting.

We derive large-mode asymptotic expansions for the eigenvalues and eigenfunction norms of the Chen-Libby perturbation problem about Falkner-Skan boundary-layer profiles. The analysis extends Brown's matched-asymptotic construction for the Blasius problem to Falkner-Skan solutions with positive wall shear – both favorable gradients and the upper branch of the mild adverse range. Although the outer and middle layers retain the same structure as in the Blasius case, the wall layer changes qualitatively when the pressure-gradient parameter $\beta$ is nonzero: the Falkner-Skan wall expansion singularly perturbs the inner Bessel problem at relative order $\Lambda^{-1/3}$, where $\Lambda$ is the large eigenvalue parameter. This produces a new wall-induced contribution $s^{-1/3}$ (where $s=n-1$ and $n$ is the large eigenvalue index) to the large-mode eigenvalue expansion, ahead of Brown's $s^{-1/2}$ correction. The new term vanishes in the Blasius limit, where Brown's ordering is recovered. The matched eigenfunction also yields asymptotic estimates for the Chen-Libby norms. The eigenvalue and norm formulae are compared with direct numerical shooting calculations for $\beta=1/2$, showing the expected asymptotic convergence and confirming the role of the new wall correction.

This article is devoted to the inverse source problem of uniquely determining a measure-valued source from sparse boundary measurements. The measurements considered consist of flux observations over a time interval at two distinct points on the boundary of the domain. The main objective of this work is to extend the existing literature on inverse source problems from sparse boundary measurements, which has so far been limited to point sources or L2 sources, to the identification of a general class of Radon measures. Our approach combines several analytical tools, including regularity properties, boundary representations, and the time analyticity of solutions to the diffusion equation with singular sources. Our theoretical analysis is complemented by a numerical study of the problem. In particular, we investigate the reconstruction of point sources and of a source supported on a curve, and present numerical experiments illustrating the recovery of such sources from sparse boundary flux measurements.

We prove that, for a given $C^{r,\beta}$-regular Riemann metric posed on a $d$-dimensional domain, every short immersion into the Euclidean space $\mathbb{R}^{d+k}$, can be uniformly approximated by exact isometric immersions of regularity $C^{1,\alpha}$ for any $\alpha<\alpha_0=\min\{\frac{r+\beta}{2}, \frac{1}{1+d(d+1)/k}\}$. Our theorem recovers several previously known results as special cases. The novelty thereof lies in providing a unified flexibility statement for arbitrary dimensions $d$ and codimensions $k$, while also treating the so far uncharted range $k\in (1, \frac{d(d+1)}{2}-d+1)\setminus \{d\}$, where no corresponding general result was previously available. Our threshold flexibility exponent $\alpha_0$ agrees with that previously obtained for the closely related Monge-Ampère system. As an application, we prove a new estimate in the quantitative immersability of thin prestrained films, setting the scaling exponent of the infimum of non-Euclidean energies in presence of an arbitrary prestrain metric, and in the limit of the film's vanishing thickness, at $\frac{4\alpha_0}{\alpha_0+1}$.

The length-constrained cycle partition problem (LCCP) is a graph optimization problem in which a set of nodes must be partitioned into a minimum number of cycles. Every node is associated with a critical time and the length of every cycle must not exceed the critical time of any node in the cycle. We formulate LCCP as a set partitioning model and solve it using an exact branch-price-and-cut approach. Our dynamic programming-based pricing algorithm to generate improving cycles exploits the particular structure of the pricing problem for efficient bidirectional search and symmetry breaking. Computational results show that the LP relaxation of the set partitioning model produces very strong dual bounds and our branch-price-and-cut method improves significantly over the state of the art. It is able to solve previously solved instances in a fraction of the time and closes 14 previously unsolved instances with numerically safe bounds, one of which has 76 nodes, a notable improvement over the previous limit of 52 nodes.

We introduce a Monte-Carlo Tree Search solver for SAT and XSAT that pairs WalkSAT-style rollouts with a two-thread symmetry-breaking initialisation: one thread starts from all-true, the other from all-false, bounding each thread's initial Hamming distance to a satisfying assignment by $\lfloor n/2 \rfloor$. Empirically the solver closes 100/100 SATLIB graph-colouring encodings (flat200-479, sw100) in tens of milliseconds each, 20/20 planted 3-XOR-SAT at $n{=}200$ (median 10~s), 6/6 at $n{=}300$ (median 138~s), and one SAT Competition 2025 instance (Break-triple-04-06.xml.cnf) in 50~ms via the polarity split alone. On a full 16-round DES key-recovery encoding ($n{=}1976$, $m{=}30072$) it drives the negative-clause count from $\sim 200$ down to 24 (99.9% clauses satisfied) over 7 hours before hitting the S-box plateau.

We introduce a class of \(q\)-deformed fractional order maps by replacing the classical binomial memory kernel in discrete fractional dynamics with Gaussian (\(q\)-) binomial coefficients. The proposed framework interpolates between memoryless discrete maps and classical fractional order maps, unifying circle- and cardioid-shaped stability regions through intermediate crescent geometries. Using the \(Z\)-transform and the \(q\)-binomial theorem, we derive characteristic equations and determine the associated stability regions in the complex plane. We further analyze the asymptotic behavior of the memory kernels, showing that the classical fractional kernel exhibits power-law decay, whereas the \(q\)- and \((p,q)\)-deformed kernels exhibit exponential-type localization. The theory is extended to nonlinear logistic-type maps and to a broader \((p,q)\)-deformed framework, where the regime \(p>q\) yields decaying memory kernels and stable dynamics. Numerical simulations illustrate the theoretical results and the interplay between the deformation parameters, memory effects, and stability geometry.

Can a model be identified from one experiment? We study analytic systems that are linearly parameterized by a combination of prescribed dictionary terms, such as partial differential operators and dynamical systems. For a single input-response pair, recovery is possible exactly when the evaluated dictionary terms are linearly independent. We prove a sharp zero-one law: either no input uniquely determines the coefficients, or almost every random input sampled from a nondegenerate Gaussian measure does. This dichotomy reduces one-shot system identification to a question about degenerate inputs and provides an a posteriori certificate for any recovered model. Numerical examples recover dynamical systems, nonlinear partial differential equations, and structured matrix families from single trajectory data, while also detecting when an extra probe is necessary.

It is known that the operator specification property does not coincide, in general, with other notions of chaos in linear dynamics. In this paper, we show that, for a natural and widely studied class of linear operators, namely the dissipative composition operators of bounded distortion (including the class of weighted shifts), specification-type properties do not introduce new dynamical behaviors: they coincide with classical notions of chaos. We also extend these equivalences with conditions on periodic points and examine in detail the special case of the translation operator $T_{\alpha}$ acting on $L^p(\mathbb R,\mathcal B,\mu)$, where the measure $\mu$ is induced by a density function. Moreover, we show that, in general, there is no relationship between the operator specification property and the shadowing property in the linear context. Finally, we analyse the important role played by the bounded distortion hypothesis.

We prove a fractal version of Lieb's Hardy-Littlewood-Sobolev inequality for orthonormal functions. On the one hand, this can be viewed as a trace theorem for orthonormal functions. On the other, it allows us to recover the Rozenblum-Tashchiyan bound for the number of negative eigenvalues of $-\Delta-\mu$, where $\mu$ is a shell potential. We also recover Rozenblum's bound for the sum of negative eigenvalues via a Lieb-Thirring kinetic inequality. Our proof is direct, avoiding both Schatten classes and variational arguments. We first reprove Adams' fractal Hardy-Littlewood-Sobolev inequality (for single functions) via Fourier analysis. This yields the required endpoint estimate as well as a bound for the interaction energy of Frostman measures.

We prove that the principal $3$-blocks of the Mathieu group ${\sf M_{12}}$ and the special linear group $\SL_3(3)$ are splendidly Rickard equivalent, and hence derived equivalent, by applying Rickard's theorem whose origin goes back to the second author. As a byproduct we answer to a question on the first Hochschild cohomology of the principal $3$-blocks above posed by W. Murphy.

Let $T$ be a tree. For a vertex $v\in V(T)$, the eccentric subtree number $\epsilon_T(v)$ is defined as $\epsilon_T(v)=\min\{f_T(v,u): u\in V(T)\}$ where $f_T(v,u)$ denotes the number of subtrees of $T$ containing both $v$ and $u$. A core vertex of $T$ is a vertex with the maximum eccentric subtree number, and the set of all the core vertices of $T$ is called the core center of $T$. The core center of $T$ consists of either a single vertex or two adjacent vertices. There are other central concepts in a tree, such as the center, centroid, subtree core, and characteristic center, and these may all be different. By $d_T(C, \mathfrak{C})$, $d_T(C_d, \mathfrak{C})$ and $d_T(S_c, \mathfrak{C})$ we mean the distance between center and core center, distance between centroid and core center and distance between subtree core and core center in $T$, respectively. We show that for any tree $T$ on $n\geq 6$ vertices, (i)]$d_T(C,\mathfrak{C})\leq \lfloor \frac{n-g_0-4}{2} \rfloor$; (ii)]$d_T(C_d,\mathfrak{C})\leq \lfloor \frac{n-5}{2} \rfloor$; (iii)] $d_T(S_c,\mathfrak{C})\leq\left\{ \begin{array}{ll} 1, &\text{if $n=7$,} n-g_0-3, &\text{if $n\neq 7$;}\\ \end{array} \right.$ where $g_0\geq 2$ be the smallest positive integer such that $2^{g_0-1}+g_0\geq n-3$. Moreover, we show that these bounds are best possible by obtaining a tree which attains these bounds. We also obtain a tree which maximizes the distance between characteristic center and core center over all trees on $n\geq 6$ vertices. The asymptotic behaviour of all these distances are also studied.

We study aggregation of statistical evidence under unknown and potentially complex dependence using group-invariance. Building on permutation-based constructions that treat transformed datasets as exchangeable units, we aggregate evidence across statistics for each transformed dataset and calibrate the resulting aggregates across transformations. We develop a finite-sample power and adaptivity theory for this framework, together with extensions to sequential and data-dependent aggregation that preserve validity. For single-batch aggregation, which uses one collection of transformed datasets for both standardization and calibration, we show that the critical values uniformly improve on deterministic calibrations valid under arbitrary dependence, including Bonferroni correction, while adapting to the unknown dependence structure. We also introduce a sequential alpha-spending version that permits early rejection when evidence is strong, and a two-batch extension that separates standardization from calibration to accommodate learned aggregation rules and reduce computation. Applications to adaptive nonparametric testing and conformal prediction illustrate how these results sharpen existing aggregation methods.

Let $p=3d+1$ be prime, let $G(p,3)=C_p\rtimes C_3$, and put $\Lambda=\mathbb{Z}[G(p,3)]$. We study tensor products of $\Lambda$-lattices representing syzygies and generalised syzygies of the trivial module. For $i=1,\,2,\,3$, we prove that $K(3)\otimes R(i)\cong R(i)\oplus\Lambda^{2d}$ and $K(3)\otimes K(i)\cong K(i)\oplus\Lambda^{2(p-d)}$. Thus, $K(3)$ acts as an identity on the corresponding stable classes under tensor product. We then construct explicit lattices $L$ and $Y$, representing dual stable classes associated with $K(1)$ and $K(2)$, and determine decompositions of several tensor products involving $R(1)$, $L$, and $Y$. As an application, we show that the stable isomorphism $L\sim Y^{\ast}$ is sufficient for $G(p,3)$ to have Wall's $\mathcal{D}(2)$-property and, under the same hypothesis, construct an associated six-periodic free resolution. Finally, we show that Johnson's ideal class injectivity condition implies $L\sim Y^{\ast}$, and formulate the remaining stable-isomorphism problem as a concrete integral-representation calculation.

We address the problem of optimally placing a limited number of sensors to reconstruct high-dimensional signals without knowledge of the underlying dynamics. The task is formulated as a nonconvex combinatorial optimisation problem and recast as a weakly convex constrained projection problem. This reformulation allows us to compute $\varepsilon$-global solutions using the Inexact Cutting Sphere algorithm. We further propose the Inverse Cutting Sphere algorithm, which starts from any feasible heuristic solution and either improves it by a prescribed tolerance $\varepsilon$ or certifies its $\varepsilon$-global optimality. The framework is evaluated on pressure reconstruction for NACA airfoils using XFOIL data.

We compute the heterotic moduli-space metric through $\alpha'^2$ for backgrounds admitting a smooth $\alpha'\to0$ limit. The Kaehler potential is unchanged when written in an appropriate field basis suggested by the ten-dimensional supersymmetry equations, while the metric receives $\alpha'^2$ corrections. We discuss this and clarify the roles of the extension bundle, F-terms, the D-terms and the string-derived moduli space metric.

We construct quantitative almost laws for $SO(3)$. More precisely, there exist a constant $c>0$ and non-trivial words $W_n\in F_2$ such that, for every $A,B\in SO(3)$, \[ \|W_n(A,B)-I\| \le \exp\!\left(-c |W_n|^{\delta}\right), \] where $\delta=\log_2(x_0)=0.879146\ldots$ and $x_0>1$ is the real root of $x^3=x^2+x+1$. This improves the exponent $\log_2\varphi$ obtained from Elkasapy's lower-central-series construction. As an application, we show how this result improves the word-length threshold in Kuperberg's Solovay–Kitaev algorithm for single-qubit gates.

In this paper, we construct a family of solutions to the $3$-fold Hull-Strominger system using the Aeppli class, without introducing the auxiliary gauge connection on the tangent bundle. In particular, we deform the conformally balanced metric along an Aeppli class off-shell and then tune it by a hermitian $(1,1)$-form dependent on this Aeppli class on-shell to satisfy the anomaly cancellation condition. The existence of the family of solutions is then obtained by the implicit function theorem. This refines the previous work by not introducing auxiliary gauge connection, thereby matching the expected dimension with the Bott-Chern parameter. This construction of the Aeppli parameter also extends to the $n$-fold Hull-Strominger system.

We present frequency-domain models of a controlled mechanical system consisting of a flexible beam and a rigid body. The transfer function for the beam, governed by the Timoshenko equations under the body-beam interface conditions, is derived analytically. Based on this infinite-dimensional representation, reduced-order models are constructed within the framework of Loewner matrices. A comparative analysis of the Bode plots is presented for the beam model using different choices of sensor-actuator pairs to validate the proposed frequency-domain modeling framework.

This paper solves the targeting problem focusing on accuracy, computational efficiency, and reliability. The trajectory optimization problem is first recast as a polynomial optimization problem (POP) by leveraging differential algebra to compute high-order Taylor expansions of the nonlinear dynamics and constraints. Moment-sum-of-squares (SOS) optimization is then utilized to solve this POP. A convex formulation based on a second-order expansion of the dynamics is also proposed. For impulsive targeting, the moment-SOS and convex approaches are compared against traditional nonlinear programming (NLP) solvers and map inversion techniques. Results indicate that the moment-SOS approach provides solutions as accurate as traditional NLP, but with the critical advantage of guaranteeing convergence to the global optimum under mild assumptions. Furthermore, the method excels at handling large maneuvers and long propagation times, conditions in which standard linear approximations rapidly degrade. To demonstrate its versatility, the methodology is extended to a continuous low-thrust station keeping (SK) scenario in the Earth-Moon Circular Restricted Three-Body Problem. The algorithm's performance is then evaluated in the presence of significant state errors. The ability to directly handle non-convex constraints and recast complex, nonlinear dynamics into formulations with reliable convergence properties makes the moment-SOS approach suitable for autonomous onboard applications.

We develop a GNS-based construction of geometric tensors on smooth parametric statistical models over $C^*$-algebras. Since the state space of a $C^*$-algebra is generally not a smooth manifold, the construction does not rely on pulling back tensors from an ambient state manifold. Instead, the GNS Hilbert spaces and their duals are organized into non-locally-trivial Hilbert fibrations over the state space. For models satisfying a compatibility condition expressing derivatives of expectation values as continuous functionals on the realified GNS fibers, each tangent vector admits a canonical dual GNS representative. Pulling back the dual GNS Hermitian product along the corresponding canonical lift produces a Hermitian tensor $K$ on the complexified tangent bundle of the model, whose real and imaginary parts define, under suitable regularity assumptions, a smooth weak Riemannian metric tensor $G$ and a smooth two-form $\Omega$. In finite-dimensional parameter manifolds the metric is, of course, strong. The construction recovers the Fisher–Rao metric in the commutative dominated case, the Fubini–Study geometry for pure states up to the normalization and sign convention imposed by the dual GNS pairing, and the SLD metric for faithful quantum states. In finite-dimensional faithful models, the two-form $\Omega$ is proportional, up to convention, to the expected commutator of the SLD representatives, equivalently to the mean Uhlmann curvature. We show through faithful qubits and displaced thermal states that $\Omega$ need not be closed. For bundle-regular models, the associated fiberwise symplectic form on the real dual GNS bundle admits connection-dependent closed extensions to the total space, while closedness of $\Omega$ on the parameter manifold is controlled by the covariant exterior derivative of the canonical real dual GNS lift.

We establish sharp interior regularity for solutions to vectorial, nonlinear local-nonlocal systems.

We study the relation between the pants graph of a non-orientable surface and that of its orientable double cover. Given a non-orientable surface $N$ with orientable double cover $\pi\colon S\to N$, we construct a natural map between pants graphs induced by lifting pants decompositions. We prove that this map defines a quasi-isometric embedding of $\mathrm{Pants}(N)$ into $\mathrm{Pants}(S)$. Using Brock's quasi-isometry between $\mathrm{Pants}(S)$ and Teichmüller space $\mathrm{Teich}(S)$ endowed with the Weil-Petersson metric, together with the identification of the Teichmüller space of $N$ with the fixed-point locus of the deck involution on $\mathrm{Teich}(S)$, we prove that $\mathrm{Pants}(N)$ is quasi-isometric to the Teichmüller space of $N$ equipped with the induced Weil-Petersson metric.

This paper studies the giant component of the sparse random bipartite graph $G(n, n, p)$, where $p = c/n$ for a fixed constant $c > 1$. We prove that its size is asymptotically normal.

We prove that the filtered Varchenko–Gelfand algebra is invariant under a natural Cremona operation on a class of real hyperplane arrangements. Namely, suppose that an arrangement contains all coordinate hyperplanes and that every remaining defining form is supported on two coordinates. Swapping the two coefficients in each such form produces its Cremona transform. Coordinatewise inversion gives a chamber bijection and an isomorphism of the corresponding filtered Varchenko–Gelfand algebras over every commutative coefficient ring. As an application, we exhibit two arrangements of eight central planes in $\mathbb{R}^3$ with isomorphic filtered Varchenko–Gelfand algebras but non-isomorphic tope graphs. This disproves a conjecture of Yagi–Yoshinaga on reconstructing tope graphs from filtered Varchenko–Gelfand algebras.

Deep Neural Networks (DNNs) used in safety-critical applications are vulnerable to hardware and memory faults that corrupt network weights and degrade reliability. In this paper, we propose a Center of Gravity (CoG) guided weight correction method that restores faulty weights based on their spatial characteristics within each layer. The proposed approach detects and corrects weight faults using distance-aware correction rules, eliminating the need for retraining or architectural modification. The effectiveness of the proposed method in terms of the capability of tolerating hardware faults has been evaluated through performing fault injection at different Bit Error Rates (BERs). Experiments on safety-critical LSTM-based Networks, including StageNet for disease progression tracking and MTFNet for cardiac anomaly detection, demonstrate fault tolerance improvements of up to 230x and 6.41x, respectively, at a BER of 10^{-3}, with negligible accuracy loss. When extended to Convolutional Neural Networks (CNNs), the method achieves up to 49.55x and 20.79x improvements under comparable fault conditions on ResNet-18 and VGG-16, respectively. To the best of our knowledge, this is the first work to apply the CoG concept to neural network weight tensors for enhancing model reliability.

This work introduces Physics-Informed Splines (PI-Splines), a structured spline-based architecture for physics-informed learning. Instead of representing the solution of a differential equation with a neural network, PI-Splines directly parametrize the unknown field through a tensor-product B-spline expansion with trainable control coefficients. This formulation preserves the residual-based training paradigm of Physics-Informed Neural Networks while providing compact support, explicit smoothness control, analytical derivatives, and a direct geometric interpretation of the trainable parameters. When compatible with the spline representation, boundary conditions can be imposed strongly by fixing suitable boundary control coefficients. The proposed method is evaluated on several benchmark problems of increasing difficulty and compared with standard physics-informed frameworks under matched governing equations, collocation sets, loss terms, and optimization procedures, so as to isolate the effect of the approximation architecture. Numerical experiments show that PI-Splines provide a competitive and stable alternative to neural physics-informed architectures, particularly in settings where structured representations, locality, and parameter efficiency are desirable.