Papers

In high-mobility doubly dispersive links, basis expansion models (BEMs) reduce channel dimensionality, yet unknown payload symbols generally contaminate conventional matched-pilot channel estimates. This paper establishes the exact conditions under which such estimates become payload-independent and derives a pilot, guard, and data-placement rule that guarantees these conditions. We prove a necessary-and-sufficient zero pilot–data interference (ZPDI) condition under which the matched-pilot least-squares (LS) solution coincides with the maximum-likelihood (ML) estimator for the reduced pilot statistic. When ZPDI holds, estimation requires a single precomputed projection. When it does not, the estimate contains a deterministic, channel-scaled payload bias that persists at high signal-to-noise ratio. A disjoint-support rule, independent of the selected basis, realizes ZPDI through pilot, guard, and data placement. We then specialize the framework to orthogonal time–frequency space (OTFS) and examine its structural and performance consequences. With the generalized complex-exponential BEM (GCE-BEM), the ZPDI estimator remains within about $2$~dB of the perfect channel state information benchmark in bit error rate at speeds up to 500~km/h.

This text is chapter 149 of the ZAG Handbook of Algebraic Geometry. We review the main results of the paper "Counting divisorial contractions with centre a $cA_n$-singularity", published in Publications of the Research Institute for Mathematical Sciences, Kyoto University.

This report studies and compares four families of Model Predictive Control (MPC) algorithms for autonomous spacecraft rendezvous guidance: Linear MPC, Tube MPC, Fast/Embedded MPC, and Successive Convexification (SCvx). Using the Clohessy-Wiltshire-Hill (CWH) relative-motion model, we show that the marginal stability of the underlying dynamics causes the condition number of the condensed Hessian to grow sharply with the prediction horizon, which explains why gradient-based solvers such as Fast MPC diverge under long-horizon stress tests while exact linear-algebra solvers (ADMM with Cholesky factorisation) remain unaffected. Tube MPC is validated across five standard rendezvous manoeuvres (Translation, R-bar, V-bar, Natural Motion Circumnavigation, and Corkscrew) using a single fixed controller configuration, and a two-sided, provably correct bound is derived to bracket the worst-case tracking error directly from the cost weight matrices. The framework is further extended to track an arbitrary, non-closed-form reference trajectory through online-recomputed linearisation, with the safety guarantee holding throughout. Finally, to assess feasibility for real flight software, the core numerical routines (Cholesky factorisation and the Riccati equation solver) are re-implemented from first principles, without external libraries, and validated against standard scientific computing tools to machine precision. Taken together, these results support Tube MPC as a robust and computationally realistic controller for onboard, real-time spacecraft rendezvous guidance.

We determine the Morse index and normal nullity of all constant-curvature minimal two-spheres in complex projective spaces, identify the normal Jacobi kernel as an $\SU(2)$-module, and prove that every normal Jacobi field is integrable. The formulas include all totally geodesic extensions. In a fixed ambient space, the normal nullity is constant along each Veronese sequence.

Triangulate a simple quadrilateral by connecting all vertices to an additional point. If the vertices carry values, the piecewise linear function can be assigned a Dirichlet energy. We show that the minimal Dirichlet energy as a function of the location of the inserted point is convex, and the location of the minimum is independent of the values at the corners - a quadrilateral has a discrete harmonic center, characterized by an equilibrium of currents across the inserted edges. It turns out that the fixed points of the Möbius involution swapping opposite corners of the quadrilateral are critical points of this energy, so the discrete harmonic center is Möbius-covariant. For tangential and cyclic quadrilaterals the center admits simple closed forms related to the circle centers. The center and its data-independence generalize to polytopes with d + 2 vertices in dimension d, but the conformal characterizations are special to four points in the plane.

The current paper considers the problem of recovering a vector-function on $[-1,1]$ from a limited number of coefficients of its expansion into a series of eigenfunctions of a one-dimensional Dirac system. The Krylov–Lanczos–Eckhoff–Gottlieb acceleration method is examined in the situation when the boundary values it requires have to be computed from the generalized Fourier coefficients themselves. This leads to a $2q\times 2q$ linear system whose matrix is a block Vandermonde matrix; its determinant and inverse are computed explicitly, and the asymptotic $L_2$-error constant of the method is found, paralleling the classical trigonometric case.

Factorization in rings of the form $K((G^{\le 0} )) $ exhibits pathological behavior due to divisibility by monomials. It remains open whether this is the only obstruction and whether the quotient by the ideal generated by all monomials is a unique factorization domain. Prior to this work, the only instances of unique factorization beyond the irreducible elements followed directly from the primality of elements which aren't divisible by any monomial and whose support has order type $\omega$ or $\omega+1$. Using the $RV$ tool from valuation theory as a key ingredient, we establish unique factorization for elements of $ K((\mathbb{R}^{\le 0}))$ which aren't divisible by any monomial and whose support has order type $\omega^{\alpha} $ or $\omega^{\alpha}+1 $ for a large class of non additively principal ordinals $\alpha$.

Coarse quantization in massive multiple-input multiple-output (MIMO) systems reduces power but causes clipping distortions. The Bayesian Expectation-Maximization (BEM) algorithm can recover clipped signals, but its matrix inversion and dynamic step-size evaluation are hardware bottlenecks. We propose a hardware-friendly one-step correction that uses the initial Jacobi-preconditioned Conjugate Gradient (PCG) direction with a fixed relaxation parameter. The resulting symbol-level update has an ultra-lightweight $\mathcal{O}(U)$ feed-forward datapath and approaches high-resolution reference detectors in the evaluated massive-MIMO setting. Our finite-dimensional analysis establishes the exact one-step descent law, proves that Jacobi normalization cancels the raw multiplicative near-far scaling while confining the loaded-system dependence to bounded attenuation factors, and gives verifiable sufficient conditions for fixed-step descent in terms of normalized channel coherence. System-level results indicate projected power savings for energy-efficient massive MIMO uplinks.

Let $\Cvec$ be a fixed orientation of the cycle $C_\ell$, $\ell\ge3$, which is not directed. For an oriented graph $D$, let $d_D^*(v):=\max\{d_D^+(v),d_D^-(v)\},$ and let \[ \sigore(D):=\min\bigl\{d_D^*(x)+d_D^*(y):x\ne y,\ xy,yx\notin A(D)\bigr\}, \] with $\sigore(D)=\infty$ if the underlying graph of $D$ is complete. We prove that, for every $\mu>0$, there exist $\gamma>0$ and $n_0$ such that every $n\ge n_0$ with $\ell\mid n$ and every $n$-vertex oriented graph $D$ satisfying \[ \alpha(D)\le\gamma n \text{ and } {\sigore(D)\ge\left(\frac34+\mu\right)n} \] contains a $\Cvec$-factor. {Additionally, for every fixed $s\ge2$ and every fixed real constant $C$, we construct arbitrarily large oriented graphs with $\sigore(D)\ge \frac34n+C$ that contain no $C_{2s}^{\ad}$-factor. More precisely, $C:=\frac34\alpha(D)-2$ for $s=2$ and $C:=\frac14\alpha(D)-\frac32$ for $s\ge3$.} This paper develops a weighted reduction framework adapted to dominant degree condition, proves the absorption lemma via closed-cluster merging with even-walk, and derives almost-perfect tiling structures by virtue of Farkas-lemma-based fractional decomposition.

We formulate the type-I seesaw as a relativistic fermionic proximity effect in the exact active-sector two-point kernel. Active neutrinos have no renormalizable gauge-invariant Majorana mass and couple by lepton-number-conserving Yukawa hybridization to a sterile Majorana pairing kernel. Integrating out the sterile fields gives a nonlocal Nambu-Gorkov embedding self-energy. In a sterile Takagi basis its anomalous and normal flavor responses are $\mathcal{A}(Q^2)=\sum_i M_i y_i y_i^T/(Q^2+M_i^2)$ and $\mathcal{B}(Q^2)=\sum_i y_i y_i^\dagger/(Q^2+M_i^2)$, respectively. Their low-energy expansions generate correlated lepton-number-violating and lepton-number-conserving moment towers. The leading moments reproduce the Weinberg coefficient and the dimension-six kinetic operator. Both towers share one sterile mass set; its positive normal support supplies a common annihilator, although anomalous residues can cancel at exact degeneracy. We prove normal block-Hankel positivity and positivity of a dimensionally rescaled mixed Nambu-Hankel sequence. The latter gives matrix Cauchy-Schwarz inequalities between lepton-number-conserving and lepton-number-violating data and can reveal paired directions hidden by a degenerate normal residue. Isolated nondegenerate pole residues satisfy an additional nonlinear compatibility identity. These results yield tree-level EFT consistency tests and a matrix-pencil reconstruction of visible pole and aggregate-residue data; selecting a unique ultraviolet Lagrangian requires additional information about sterile-basis multiplicities. An exact one-generation example distinguishes the static Schur-complement kernel from the physical Takagi pole mass. The proximity structure applies without a Fermi surface or condensate, and Nambu doubling introduces no new physical degrees of freedom.

Many statements studied in reverse mathematics can be seen as mathematical problems, formulated in terms of instances and solutions. We develop a framework of typicality encompassing measure and genericity, and we classify the reverse mathematics zoo in terms of which problems admit typical solutions. It turns out that even very weak problems do not admit typical solutions.

This paper studies partially observed mean field games without perfect recall (WPR). The representative agent observes a noisy signal, but the control at time \(t\) uses only \(\mathcal G_t^I=\sigma(y_t)\), a generally non-nested information family. The conditional population law instead uses the observation filtration \(\mathbb F^Y\). These coupled levels rely on different information scales and are difficult to close within one construction. We parameterize the environment by a deterministic compatible joint law of state, driving variables, and random mean field term, thereby preserving its dependence structure without enlarging the agent's control information. For a fixed law, a reference measure and Girsanov's theorem yield a WPR stochastic maximum principle; the selected response is represented by the conditional Hamiltonian and WPR belief measure. The joint path posterior of hidden state and mean field term gives a weak Kushner-Stratonovich representation of the conditional population law. A recursive response map is continuous on a compact convex set of compatible laws, so Schauder-Tychonoff yields a weak WPR equilibrium. For a fixed equilibrium law and feedback, a compatible Yamada-Watanabe theorem lifts pathwise uniqueness to a strong realization. Finally, a linear-quadratic interbank lending example compares perfect recall (PR) with WPR. The PR response follows the Kalman-Bucy feedback, whereas the WPR response solves a Fredholm-Volterra equation and is affine in the current observation under Gaussianity. The numerical experiment illustrates how equilibrium behavior differs between PR and WPR.

In this paper, we propose a novel objective function-free trust-region method designed to solve optimization problems over decentralized networks. Unlike traditional approaches that often rely on stepsize tuning, our framework employs a function-free trust-region procedure that enables adaptive selection of the step length. Our approach accommodates first- and second-order models and eliminates the need to share local function values and gradients among agents, thereby enhancing privacy and computational efficiency. On the theoretical side, we establish provable iteration complexity guarantees that, for some variants, match those established for classical centralized trust-region methods. Numerical evaluations demonstrate that our approach achieves a favorable trade-off between performance and efficiency, requiring only moderate communication overhead compared to state-of-the-art methods in the literature.

Non-collision singularities of the $n$-body problem are initial conditions for which no global solution exists and which, however, do not lead to a collision in the limit; i.e., the moment of inertia of the system diverges. The question of whether the set of initial conditions leading to non-collision singularities in the $n$-body problem is improbable is open and is the first on Barry Simon's list of fifteen problems in mathematical physics from the year 1984. So far, this question has only been answered positively in the case $n=4$. Using a final cluster decomposition into subsystems whose particles interact strongly with each other and exert only small forces on particles from different subsystems, we can prove the improbability of the set of all singular orbits with the following conditions: -There may be arbitrarily many subsystems undergoing total collision at their center of mass. - There may be subsystems exhibiting a non-collision singularity, i.e. the respective moment of inertia of the subsystem diverges. These subsystems consist of exactly four particles each and the asymptotic directions of these subsystems have to differ (which is only a mild condition). This result includes all known statements and indicates a way how the general problem could be solved.

We establish sharp mixed spectral Barron regularity for eigenfunctions of molecular Coulomb Hamiltonians. The mixed norm is a Fourier $L^1$ norm with one isotropic weight and coordinate-product weights, and therefore detects regularity invisible to the isotropic Barron scale. For a nonempty set $I$ of electron indices on which the wave function is antisymmetric, we derive an explicit admissible region for the isotropic order $s$ and the coordinate orders $\alpha,\beta$. This region is optimal as a uniform statement over the class of clamped-nuclei Coulomb Hamiltonians. For fixed-spin components with two occupied spin blocks, it reduces to $s+\alpha+\beta<1$; in the fully spin-polarized class it reduces to $s+\alpha<1$. In particular, if $\mathcal I_\sigma$ denotes the family of occupied same-spin blocks determined by $\sigma$, then every fixed-spin spatial component $\psi_\sigma$ satisfies, for every $0\leq\alpha<1$, \[ \left(\sum_{I\in\mathcal I_\sigma}\prod_{i\in I}\langle\xi_i\rangle^\alpha\right)\widehat{\psi_\sigma}\in L^1(\mathbb{R}^{3N}). \] For a fully spin-polarized state, $\mathcal I_\sigma=\{\{1,\ldots,N\}\}$.

Tensor networks, originally developed for quantum many-body physics, are promising models for machine learning. We derive stochastic Riemannian optimizers for tree tensor networks (TTNs) on both their parameter and quotient manifolds, including adaptive and learning-rate-free schemes suitable for minibatch training. Using a hybrid CNN-TTN architecture, we evaluate the methods on Fashion-MNIST, CIFAR10, and Imagenette. The proposed optimizers achieve predictive performance comparable to unconstrained optimization while enabling numerically stable downstream compression.

We prove uniform-in-$N$ partition-function estimates at all positive temperatures for the centered modulated energy of logarithmic, Riesz, and Bessel–Riesz kernels on $\mathbb R^d$ in the locally square-integrable range $0\le s<d/2$. They yield finite-particle exponential integrability, quadratic behavior at small parameter, explicit kernel-dependent growth at large parameter, and uniform entropy control of the associated Gibbs measures. As applications, we obtain uniform Rényi-divergence and relative-entropy bounds for interacting Gibbs equilibria and entropic mean-field convergence near thermal equilibrium; the optimal $N^{-1}$ normalized relative-entropy rate for the three-dimensional Coulomb flow; and an $N^{-1/2}$ Gaussian approximation for fixed-time finite-dimensional fluctuations. Finally, the modulated energy converges to a random variable in the second Wiener chaos, and for every positive parameter the partition functions converge to its Laplace transform, represented by a Carleman–Fredholm determinant. This formula identifies the sharp small-parameter behavior and, when the reference density is bounded below on a ball, the sharp large-parameter growth.

Low-thrust trajectory optimization is a central task in interplanetary mission design, but its nonlinear dynamics and operational constraints often lead to challenging non-convex optimal-control problems. Sequential convex programming has emerged as an effective approach to address these problems, while quantum annealing offers a complementary paradigm for solving quadratic unconstrained binary optimization problems. This paper introduces quSCP, a quantum-based sequential convex programming framework that reformulates each convex subproblem as a quadratic unconstrained binary optimization problem suitable for quantum and hybrid quantum–classical solvers. Equality, inequality, and trust-region constraints are embedded through quadratic penalty terms, while an iterative refinement strategy is used to reduce the accuracy loss introduced by binary discretization. The method is assessed on a fuel-optimal Earth–Mars low-thrust transfer by comparing standard sequential convex programming, a continuous quadratic unconstrained formulation, direct quantum processing unit sampling, and D-Wave hybrid solvers. Results show that quSCP produces physically consistent trajectories with propellant consumption and nonlinear constraint violations close to classical benchmarks. Direct quantum annealing is feasible only for small instances because of embedding overhead and hardware connectivity limits, whereas hybrid solvers scale to larger discretizations. Although no computational quantum advantage is demonstrated with current hardware, the results show that quantum and hybrid quantum–classical optimization can already provide competitive solutions for demanding trajectory design problems.

Let $R\ltimes_{n}M$ be the $n$-trivial extension of a ring $R$ by an $R$-bimodule $M$. We first characterize projective, injective, flat and finitely generated modules over $R\ltimes_{n}M$. Then we describe when the ring $R\ltimes_{n}M$ is left perfect (Noetherian, coherent, Artinian, hereditary, Kasch). Finally, we establish some homological formulas of $n$-trivial extensions of rings.

We assess computed lift coefficient data for flow past a circular cylinder to evaluate their suitability for practical applications. Specifically, we consider lift coefficient data for a circular cylinder over Reynolds numbers from 120 to 8000. The results are obtained from two-dimensional finite element simulations of the incompressible Navier-Stokes equations using pressure robust discretizations. We compare the computed lift coefficients with published experimental and numerical results, finding good agreement in some cases but significant disagreement in others. Because lift fluctuations are central to vortex-induced vibration concepts, these data therefore provide input for the analysis and preliminary design of bladeless turbines.

Transport in certain systems fails when diffusion cannot replenish or remove material from a boundary rapidly enough, causing the boundary concentration to reach a critical minimum or maximum. Motivated by experiments in electrochemical systems that demonstrate this diffusion-limited failure of charge transport, we determine dynamic current protocols that maximize charge transfer subject to a prescribed concentration constraint. The optimal protocol has a "bang-ride" structure: the maximum feasible current is used until the boundary concentration reaches its critical value, after which the current is progressively reduced to maintain that value. For semi-infinite domains, we derive the optimal currents analytically and obtain system-independent upper bounds on their improvement over constant-current operation. We also extend the framework to finite and multilayer domains and show that the same formulation applies to both charging and discharging.

In this paper, we prove that any complete, compact or noncompact, almost-Kähler gradient shrinking Ricci soliton is Kähler in arbitrary even dimension. Among other applications, combining our result with the classification of complete gradient shrinking Kähler-Ricci solitons in complex dimension two, we obtain a full classification of complete almost-Kähler gradient shrinking Ricci solitons in real dimension four.

Let $1 \leq q \le 2$ and denote by $2 \leq q'$ its corresponding conjugate exponent. We prove the continuity of solutions $u \in W^{1,(\frac{n}{n-1},q')}(B^n, \mathbb{R}^m)$ to the critical elliptic system $-\Delta u = \Omega \cdot \nabla u$ in dimension $n \ge 3$, where the potential $\Omega \in L^{(n,q)}(B^n, \mathfrak{so}(m) \otimes \wedge^1)$ is antisymmetric. First, we construct $P \in W^{1,(n,q)}(B^n, \mathrm{SO}(m))$ such that the PDE can be rewritten as $-\operatorname{div}(P^{-1}du) = \ast d\xi \cdot P^{-1}du$, which is nearly a Jacobian structure up to the rotation $P$. Second, we provide a Rivière's $\mathrm{GL}(m)$-Gauge in order to establish a "full" $(A,B)$-conservation law, i.e. $-\operatorname{div}(Adu)=d^\ast B \cdot du$. We show that the assumption on $\Omega \in L^{(n,q)} (B^n, \mathfrak{so}(m) \otimes \wedge^1) $ for $q\leq 2$ is optimal.

This article solves the Indefinite Summation Problem (ISP) for the difference ring $(A, \alpha)$, where $A$ is the Laurent ring of shift operators on the lattice $\Z^n$, and $\alpha$ is any ring automorphism of $A$ of finite order. The space of summable elements in $(A, \alpha)$ is infinite dimensional, and it is possible to write down an explicit basis for it. The solution translates to a finite procedure involving a matrix multiplication, where the size of the matrix can be estimated. It follows that the arithmetic complexity of the solution can also be determined. These results extend to a solution of the ISP for the ring of functions on $\Z^n$, on which $\alpha$ acts by duality. The article also points out that the solution to the ISP amounts to calculating the group cohomologies $H^i([\alpha], A), i = 0, 1$, where $[\alpha]$ is the cyclic group generated by $\alpha$.

We introduce the Krasner completion of a band equipped with a decreasing filtration by subgroups of its unit group and give a criterion for reconstruction from its quotient bands. We recover the Krasner–Linzi inverse-limit theorem for valued fields. We also obtain a valuation ring analogue of the Krasner–Linzi theorem and compare Krasner completions of unit bands with adic completion.

In this paper, we establish a stability estimate and a uniform estimate for the modulus of continuity of solutions to the degenerate complex Monge-Ampère equation in big cohomology classes. Consequently, we prove the uniqueness of solutions to complex Monge-Ampère mean field equations for a sufficiently small parameter.

A long-standing open problem asks whether there exists an infinite-dimensional Banach space on which every bounded linear operator attains its norm. With $\mathrm{NA}_1(X,Y)$ and $\mathrm{NA}_2(X,Y)$ denoting the classes of operators whose adjoints and second adjoints, respectively, attain their norms, we prove that \[ \mathrm{NA}_2(c_0,c_0)=\mathcal{L}(c_0,c_0) \qquad\text{and}\qquad \mathrm{NA}_2(\ell_1,\ell_1)=\mathcal{L}(\ell_1,\ell_1), \] providing, to the best of our knowledge, the first known infinite-dimensional spaces on which every operator has a norm-attaining second adjoint. Building on the result for $c_0$, we undertake a systematic study of this equality within a natural family of $\ell_1$-preduals given by hyperplanes of $c$, obtaining a complete characterization in this setting. In particular, we prove that \[ \mathrm{NA}_2(c,c)\neq \mathcal{L}(c,c), \qquad\text{whereas}\qquad \mathrm{NA}_3(c,c)=\mathcal{L}(c,c). \] We also establish Holub–Mujica-type theorems for the classes $\mathrm{NA}_1$ and $\mathrm{NA}_2$. More precisely, under suitable separability and approximation property assumptions, the identity $\mathcal L(X,Y)=\mathrm{NA}_1(X,Y)$ forces every operator from $X$ into $Y$ to be compact, whereas $\mathcal L(X,Y)=\mathrm{NA}_2(X,Y)$ forces every weakly compact operator from $X$ into $Y$ to be compact. Finally, strengthening a construction of Ostrovskii, we show that every infinite-dimensional Banach space admits an equivalent norm and a projection whose second adjoint does not attain its norm.

We study finite-time singularities in $n$-body systems with pair potentials that are homogeneous of degree $-\alpha$, where $0<\alpha<2$, that are not necessarily attractive and with bounded time-dependent external forces. We develop a systematic method to analyse a colliding subsystem in the presence of simultaneous collision or non-collision singularities elsewhere in the system. We extend classical results of von Zeipel and Painlevé to such subsystems, establish bounds on their internal energy near total collision, and implement a McGehee blow-up that remains effective without conservation of energy. The blow-up yields the asymptotics of the cluster size, and in the case of attractive potentials, the convergence towards the set of central configurations and asymptotics of relative position, relative velocities, and internal angular momentum. As an application of the asymptotic results, we construct a simplified proof that the initial conditions leading to collision form a set of measure zero.

The Levenberg-Marquardt (LM) algorithm is a well-known second-order method for rapid convergence and strong robustness when training small- to medium-sized neural networks (NNs). However, its computational and memory costs increase significantly as the number of parameters in an NN grows. To address this limitation, subspace methods have been proposed, such as the Krylov subspace LM (KSLM) and the hybrid subspace LM (HSLM), making second-order algorithms more efficient. In this work, we evaluate the subspace Levenberg-Marquardt algorithms for regression and classification tasks in neural networks. We compare the performance of subspace LM variants with the classical LM method, as well as other popular first-order algorithms, such as stochastic gradient descent (SGD) and Adam.

The Schottky–Klein prime function is a standard tool for boundary-value problems on multiply connected circular domains. Because this function is represented as an infinite product over a Schottky group, numerical evaluation requires truncation to finitely many factors. The standard word-length truncation grows exponentially in cost and becomes inefficient when the boundary circles nearly touch one another or the unit circle. To address this difficulty, we assign to each group element a cross-ratio potential measuring the size of its contribution, and retain only terms below a prescribed threshold. We establish uniform closed-form bounds on the change in this potential when prepending Schottky-group generators, and from these bounds we derive an efficient enumeration algorithm. The resulting relative error decays exponentially with the threshold at a rate determined by the Hausdorff dimension of the limit set of the Schottky group. Numerical experiments demonstrate that the proposed formulation achieves substantial computational speedups over word-length truncation in challenging geometric configurations.