Papers

We study the Euler-Maruyama approximation of the degenerate stochastic differential equation $dX_t = |X_t|^\alpha\,dW_t$, $0 < \alpha < 1/2$, $X_0 = 0$, for which uniqueness in law fails. The Euler scheme started exactly from zero is trapped by the absorbing solution, whereas a nonzero initialization $x_h \to 0$ can select a different weak solution. We identify the initialization-dependent limiting regimes. The critical spatial scale is $r_h = h^{1/[2(1-\alpha)]}$, and the relevant depth parameter is $\Theta_h = h\log(1+\log^+(r_h/|x_h|))$. If $\Theta_h \to 0$, the scheme converges to the non-sticky weak solution. If $\Theta_h \to \theta \in (0,\infty)$, it converges to the solution that stays at zero for the deterministic time $\theta/|\log\alpha|$ and then evolves as the non-sticky solution. If $\Theta_h \to \infty$, the scheme converges to the absorbing solution. The proof is based on an exact scale-invariant Euler chain, a logarithmic exit-time law of large numbers, a discrete Tanaka occupation estimate, and a compact-uniform restart argument. Every finite deterministic delay can be obtained by a suitable vanishing initialization, while oscillating depth parameters yield process-level nonconvergence. Thus a vanishing initialization can survive in the limit as a macroscopic deterministic delay and thereby act as a solution selector for the Euler-Maruyama approximation.

Variance reduction improves the convergence rate of stochastic non-convex optimization under the average smoothness condition. Existing adaptive variance reduction methods are especially designed for average smoothness and do not adapt to standard smoothness. In this paper, we study how one algorithm can adapt to these two settings and the noiseless setting without knowing the smoothness constants or the variance bound. Given the gradient budget $T$, we develop a two-phase method. The first phase compares two independent gradients and switches to normalized momentum when their difference exceeds a threshold. This phase provides both a candidate with standard smooth and noiseless guarantees and also ensures a bounded expected function gap for the second variance reduction phase. The resulting rates are $O(T^{-1/3})$ under average smoothness, $O(T^{-1/4})$ under standard smoothness, and $O(T^{-1/2})$ without noise. When $T$ is unknown, we propose a method with two paths that attain the corresponding rates with logarithmic factors in the stochastic settings. Here, we use a variance reduction path to deal with average smoothness and employ the AdaGrad-Norm for standard smoothness and noiseless setting. Neither method restarts from the initial point or requires a growing number of oracle calls in each round.

Let $D$ be an oriented graph. For $X \subseteq V(D)$, the inversion of $X$ in $D$ is the oriented graph that is obtained via changing the direction of all edges which have both endpoints in $X$. The inversion number of $D$, $\operatorname{inv}(D)$, is the minimum number of inversions necessary in order to remove all cycles from $D$. A dijoin $D_1 \to D_2$ is the oriented graph obtained by taking disjoint copies of $D_1,D_2$ and adding edges from all $v \in V(D_1)$ to each $u \in V(D_2)$. It was conjectured by Bang-Jensen, da Silva and Havet that $\operatorname{inv}(D_1 \to D_2) = \operatorname{inv}(D_1) + \operatorname{inv}(D_2)$ for all $D_1, D_2$. This was disproved by simultaneous constructions of various $D_1$, $D_2$ with $\operatorname{inv}(D_1 \to D_2) = \operatorname{inv}(D_1)+\operatorname{inv}(D_2)-1$. However, no known constructions achieved more than this gain of 1. Moreover, all of the constructions exploit a gap of 1 between $\operatorname{inv}(D)$ and the minimum rank of a family of related matrices, a gap that can never be more than 1, suggesting that perhaps something close to the original conjecture might be true. We prove that in fact additivity fails badly. In particular, for all $\varepsilon>0$ we show that there exist tournaments $D_1$, $D_2$ with arbitrarily large inversion numbers such that $\operatorname{inv}(D_1 \to D_2) \leq \left(\frac{1}{2} +\varepsilon\right)(\operatorname{inv}(D_1)+\operatorname{inv}(D_2))$.

We consider an inexact version of the Schwarz domain decomposition method in $3$D to solve a Laplace problem in a domain that is the union of two overlapping spheres. We solve each subproblem approximately by projecting its boundary data $L^2$-orthogonally onto the space of spherical harmonics of degree at most $N$. The model problem arises from the ddCOSMO solver used for implicit solvation in computational chemistry. We analyze the Schwarz-spherical domain decomposition method by establishing a new variant of the maximum principle.

We show that certain classes of unital, simple, separable C*-algebras with comparison are singly generated, including tracially AF algebras and monotracial crossed products of actions on the Cantor space. In particular, we show that simple, separable, tracially AF algebras have generator rank one.

Momentum methods such as STORM and second-order momentum track the gradient by adding a transport correction to momentum, at the cost of an extra oracle call per iteration. We introduce randomized momentum transport, which evaluates the correction with probability $q$ and otherwise uses a cheap prediction, with importance weighting that preserves the conditional mean and adds $(q^{-1}-1)$ times the prediction loss to the variance. Under transport-bias, second-moment, and residual-control assumptions, linear minimization oracle (LMO) updates, including normalized, sign, and exact spectral (Muon) directions, retain an $O(T^{-1/3})$ stationarity bound for each fixed $q>0$ with a single-gradient initialization. For unnormalized STORM, some $q<1$ lowers the leading coefficient of our sufficient gradient budget if and only if the gradient-difference noise modulus exceeds the smoothness constant, by at most about $10\%$. An online diagonal predictor admits a regret-to-convergence guarantee. At approximately equal oracle cost, querying a quarter of the corrections lowers the final training loss by $18$ to $43\%$ on MNIST1D and raises the test accuracy of Hessian-vector transport on CIFAR-10 by $1.5$ points.

We present new algorithms for stochastic optimization problems in which the objective is neither smooth nor convex, subject to a convex constraint set. Our algorithms return a $(\delta,\varepsilon)$-Goldstein stationary point, as recently defined for constrained problems by Liu et al. (2024), with improved complexities, and are the first to achieve optimal convergence rates for this task in several settings. In particular, we design projection-based and projection-free algorithms that achieve a $\widetilde O(\delta^{-1}\varepsilon^{-3})$ rate using stochastic gradients, which we show to be optimal, and a $\widetilde O(d\delta^{-1}\varepsilon^{-3})$ rate using zeroth-order queries, which is also optimal in its dependence on the dimension $d$ and improves over prior work. Notably, our lower bound is given in terms of the Frank-Wolfe gap and holds even for smooth problems, which is of independent interest. These results show that in nonconvex stochastic optimization, the convergence rate is degraded neither by the presence of constraints nor by nonsmoothness. Our analysis is based on a reduction to online linear optimization, extending a related technique from unconstrained optimization.

We establish that bond Hamiltonian densities, up to local difference terms whose sum vanishes under periodic boundary conditions, lying in the commutant of bond operators, $w_j\pm w_{j+1}$, is a necessary and sufficient condition for a one-dimensional nearest-neighbour Hamiltonian to conserve a uniform or staggered sum of the one-site operators $w_j$. Starting from these local commutants, we show that every such Hamiltonian and its conserved charge can be obtained from monodromy matrices constructed using regular and non-regular members of scalene Yang-Baxter triples. The results hold on periodic chains of length $L\geq3$, with even length required in the staggered case. The results do not require translation invariance, and hence the interaction on each bond can be chosen independently from the full corresponding commutant. Our representation-independent construction applies in arbitrary finite local dimension and exhausts the Hamiltonians with either prescribed additive symmetry, without requiring ordinary Yang-Baxter integrability. We illustrate the construction with spin-$\tfrac12$ and spin-1 systems and discuss its relation to the double-commutant theorem. Classifying the one-site generators $w$ and their associated bond commutants provides a route to classifying such Hamiltonians with additive conserved charges.

Multiple-gradient descent is typically formulated in Euclidean geometry, whereas matrix-aware optimizers use directions induced by other norms. We analyze a single-loop multi-objective method combining a norm-constrained linear minimization oracle with entropic updates of task weights. For any finite-dimensional norm pair, we establish an $O(T^{-1/2})$ ergodic stationarity rate in the deterministic setting, deriving the required trajectory gradient bound from smoothness and lower-boundedness in the unregularized case. With stochastic gradients and a bounded-gradient assumption, one persistent blended-gradient momentum buffer yields an $O(T^{-1/4})$ rate whose leading term has no explicit dependence on the number of tasks. We also establish static regret for the inner weighting objective on arbitrary bounded gradient sequences and, with entropy regularization and a weight floor, iteration-wise inner-gap control after burn-in while retaining ergodic stationarity, conditional on a trajectory gradient bound. The optimal blended gradient can be nonunique in spectral and sign geometries, motivating a selection-independent inner gap. Our analysis extends the spectral construction studied by concurrent MOON to general norms. Experiments compare the method with existing multi-task optimizers and examine its oracle, weighting, and momentum choices.

This paper studies pseudofinite abelian groups and counting properties of definable sets in finite abelian groups and their ultraproducts. First, we characterize pseudofinite abelian groups in terms of Szmielew invariants and, building on results of Macintyre, extend this analysis to obtain a characterization of pseudofinite $\omega$-stable abelian groups. We then give an alternative proof that the class of finite abelian groups forms a multidimensional exact class, explicitly describing the possible cardinalities of definable sets in classes of finite abelian groups. We also investigate $N$-dimensional asymptotic classes of finite abelian groups and provide a series of examples showing that, despite the exactness result, several natural classes of finite abelian groups fail to be $N$-dimensional asymptotic classes for every $N\geq 1$. Finally, in an appendix, we characterize the SU-rank of pseudofinite abelian groups, building on work of Rogers. In particular, we show that for every pseudofinite abelian group $A$, either $\operatorname{Th}(A,+)$ is superstable of finite SU-rank or it is strictly stable.

We prove scattering in $L^2(\mathbb{R}^2)$ for real-valued solutions of the Kadomtsev–Petviashvili II (KP-II) equation with localized initial data. No smallness assumption is made, no zero mean condition is imposed, and the admissible data include all Schwartz functions. The proof uses the inverse scattering transform attached to the heat operator. It rests on three ingredients: an a priori bound for the scalar $\bar\partial$ problem which depends on the size of the scattering coefficient but not on its oscillating phase, a nonlinear Plancherel identity proved by analytic continuation from small data, and an estimate showing that the nonlinear part of the reconstruction is negligible on the propagation cones of free waves. To the best of our knowledge, this is the first result establishing scattering for the KP-II equation for every real-valued Schwartz initial datum.

This work is concerned with the focusing quintic nonlinear Schrödinger equation on a star graph with two edges, with a delta interaction of arbitrary strength at the vertex. We construct a non-radial, minimal mass, finite time blow-up solution, whose singularity forms at an arbitrarily prescribed interior point of one edge. Our approach is based on a suitable cut-off of the standard pseudoconformal blow-up solution of the focusing quintic NLS on the line, inspired by Ogawa-Tsutsumi and Burq-Gérard-Tzvetkov. A noteworthy feature of our construction is the persistence of the domain of the Hamiltonian under the flow, obtained by the persistence of regularity method of Kato. We also identify a two-parameter family of distinct blow-up profiles, which should play a central role in the classification of non-radial minimal mass blow-up solutions on the star graph. Our construction is generic and can be extended to a star graph with an arbitrary number of edges. It can certainly also be adapted to other types of metric graphs.

We present a simulation-based method for the estimation, sensitivity analysis, and optimization of distortion risk measures (DRMs), a class of performance measures that capture a wide range of risk attitudes through an underlying distortion function. Existing simulation approaches often approximate DRMs using numerical partitions or empirical distributions. We show that, when the distortion function is piecewise polynomial, the nonlinear structure of a DRM can be handled exactly, without introducing distortion discretization error, using only simulated system outputs. This representation enables the development of a computational method that integrates finite-difference techniques for sensitivity analysis and for solving a broad class of differentiable risk-based optimization problems. Under suitable conditions, we establish strong consistency and convergence rates for DRM estimation, sensitivity analysis, and optimization. We further extend the approach to non-polynomial distortion functions through polynomial approximation. Numerical experiments on synthetic and calibrated financial examples demonstrate the effectiveness and robustness of the proposed approach across a range of applications.

We study the gauge of the travel time tomography problem for elliptically anisotropic planar media. Under this assumption the respective slowness surfaces consist of an inner ellipse and an outer circle. Hence, the singularities of seismic waves propagate on geodesics given by two Riemannian metrics of which the one related to shear waves is conformally Euclidean. We show that if both of these metrics are lens rigid then, up to boundary fixing diffeomorphisms, each elliptically anisotropic stiffness tensor is uniquely determined by the lens data whenever it has at least one isotropic point. If none of those points exists, then there are exactly two elliptically anisotropic stiffness tensors with the same lens data. Clearly, not all boundary fixing diffeomorphisms preserve the imposed structure of the slowness surface. We show that the set of well-behaved diffeomorphisms is never a subgroup of all boundary fixing diffeomorphisms. Our proofs are based on an interplay of algebra and geometry, and the key of the proofs is a characterization of the elliptic anisotropy via several polynomial conditions.

By establishing a weighted Lojasiewicz-Simon inequality for maps which are close to a fixed harmonic map modulo scaling, we prove Topping's conjecture on continuity of the body map in the case of finite time singularities at which only isolated bubbles form.

We study the completed Jacobi algebras of $x^3+xy^3+p(y)$ for $\operatorname{ord}_yp=n\ge4$. The explicit normal-form morsification, the filtered length and centre proofs, and the cyclic terminal identities give finite free semisimple deformations. For every parameter of finite order we construct a polynomial curve neighbourhood and a local projective small contraction; the resulting geometric GV and Donaldson–Thomas calculations identify the block multiplicities. The one-dimensional limit $E_{6,\infty}$ is studied separately as an order: we determine its normalization, conductor, radical-idealizer chain, and maximal overorders. The canonical chain yields a nine-dimensional finite quotient whose Ringel dual is isomorphic to its opposite. An expanded construction gives a choice-independent $22$-dimensional quotient and a $40$-dimensional Ringel dual; a corner quotient relates the two finite constructions. The length and centre formulas are proved by a filtered comparison with a homogeneous matrix order and by terminal identities at the exceptional endpoint.

We consider the normal form of a 1:1 detuned resonance that can be related, via a double averaging process, to the three body problem of celestial mechanics. In view of the symmetry in the gravitational coupling between the planets, the obtained normal form exhibits inversion symmetry. We provide an analytical prediction of the possible bifurcations in terms of the detunings and the other physical parameters of the system. We show that the bifurcation scenario is rich. It includes bifurcation sequences where the unstable orbit appearing after a saddle-node bifurcation can have a reversed pitchfork bifurcation, where it turns into stability with the birth of two unstable orbits, which eventually reconnect with one of the two quasi-normal modes. Furthermore, another scenario shows a standard pitchfork in which one of the quasi-normal modes becomes unstable, giving rise to a pair of stable periodic orbits. By introducing coupling and asymmetry parameters, we provide an effective way to discern all the possible sequences of bifurcations.

This paper addresses the local convergence analysis of the stabilized sequential quadratic programming method for a class of composite optimization problems with convex piecewise linear- quadratic (CPLQ) modeling functions. We demonstrate that an appropriate second-order sufficient condition (SOSC) -formulated for composite problems- alone is sufficient to guarantee local superlin- ear convergence of the primal-dual sequence generated by this method. Furthermore, we examine the relaxation of the SOSC and show that it can be replaced with a weaker noncriticality of the associated Lagrange multiplier, provided, the modeling function is polyhedral and a relative interior condition is satisfied. We also present counterexamples showing that the relative interior condition is indispensable and that this relaxation fails for general CPLQ modeling functions.

We obtain sharp isoperimetric-type upper bounds for Steklov eigenvalues on warped products whose bases are compact Riemannian manifolds with boundary and whose fibres are closed Riemannian manifolds. These bounds involve the volume of the warped product and of its boundary, as well as the eigenvalues of the Laplace operator on the fibre and the Steklov eigenvalues of the base. After separation of variables, these bounds are obtained by comparison with auxiliary Helmholtz-Steklov problems on the base involving weighted Laplace operators naturally induced by the warping function. We also study the effect of various natural geometric constraints, such as the warping function being identically equal to one on the boundary. The proofs make use of various spectral approximation methods, including homogenisation on manifolds when the fibres are one-dimensional.

As engineering systems are optimized to reduce weight and improve performance, vibration behavior, including nonlinear effects, is becoming increasingly important to analyze. While methods such as harmonic balance are popular for analyzing periodic nonlinear vibration behavior, these analyses often only consider deterministic loading. However, real applications experience additional noise in the excitation, and alternative approaches are frequently limited to smaller models. This work expands on a recent approach to calculate a time-varying covariance using a linearized system around a periodic nonlinear response. Specifically, numerical conditioning is addressed and improved for cases increasing from a few degrees of freedom to hundreds of degrees of freedom. The algorithm is then extended to capture correlations across both space and time via a period-spanning covariance matrix. Model reduction is proposed for this calculation to maintain computational efficiency. The period-spanning covariance matrix is used to sample full stochastic cycles and estimate fatigue-relevant statistics up to 1393 times faster than stochastic time integration. Importantly, the present methodology uses the same gradient information as stability analyses and continuation solutions, so it can easily be applied for many nonlinear analyses. The methodology is demonstrated for two beam systems: one with unsteady aerodynamic forces and another with a Bouc-Wen frictional tip force. For the aeroelastic model, statistics are accurately estimated via the present approach. For the frictional system, some errors are introduced compared to stochastic time integration and are attributed to the linear time-periodic nature of the system used in the present stochastic calculations. However, estimated stochastic statistics remain useful given the significant reduction in computation time.

We study the Arens regularity of the Fremlin projective tensor product of Banach lattice algebras. We firrst introduce and investigate L-weakly almost periodic functionals, a class of functionals defind in terms of L-weak compactness of their orbits. We obtain a characterization of these functionals in terms of bounded disjoint sequences and use it to analyze the Fremlin projective tensor product. In particular, we prove that, for nontrivial Banach attice algebras A and B, Arens regularity of A j jB implies Arens regularity of both A and B. The converse, however, fails: we construct two Arens regular Banach lattice algebras whose Fremlin projective tensor product is not Arens regular. We also establish several sufficient conditions for the Arens regularity of A j jB, including conditions involving the positive Schur property and order continuity of the norm on the space of regular operators. Finally, we show that when both factors are Banach f-algebras, their Fremlin projective tensor product is Arens regular.

We study the list chromatic number of generalized Johnson graphs. Recall that the vertices of such graphs are all possible $r$-element subsets of $[n] = \{1, \dotsc, n\}$ and an edge is drawn between two vertices if and only if corresponding sets have exactly $s$ common elements. We prove that for any fixed $s \geq 1$ and $r \geq 2s + 2$ we have $\chi_L(G(n,r,s)) = \Theta(n^{s+1}\ln n)$. In case where $s = 0$ we prove that for all $4\ln n \leq r \leq n/20$ we have $\chi_L(G(n,r,0)) \geq \frac{n}{120}\ln\left(\frac{en}{r}\right)$. Together with previous results of Bulankina, Inozemtsev and Kupavskii this proves that $\chi_L(G(n,r,0)) = \Theta\left(n\ln\left(\frac{en}{r}\right)\right)$ uniformly for $2 \leq r \leq (1/2-\varepsilon)n$.

We show that the cotangent bundle of a holomorphically flexible Stein manifold has the symplectic density property. As application we deduce holomorphic approximation for Hamiltonian diffeomorphisms of real cotangent bundles by complex symplectic automorphisms of the complexified cotangent bundles, which preserve the real bundle, in the Carleman sense, i.e. in the Whitney topology.

Non-Bloch band theory recovers the open-boundary spectrum of a non-Hermitian chain from its Bloch Hamiltonian evaluated along the generalised Brillouin zone rather than the unit circle. We develop its finite-size counterpart for banded Toeplitz matrices with real open-boundary spectrum. Reality of the spectrum forces the generalised Brillouin zone to be parametrised by an analytic polar curve, so the known sufficient condition for a real spectrum is also necessary under some mild non-degeneracy conditions. At finite size $n$, the eigenvalues of the original and the symmetrised Bloch Hamiltonians are given by samples at $n$ quantised momenta from the same limiting band, although separated by a phase mismatch of order $1/n$ which we obtain in closed form. Iterating this correction $K$ times yields a quantisation rule on the generalised Brillouin zone reproducing the finite spectrum with an error $\mathcal{O}(n^{-K-1})$ for any fixed $K$. At finite size, the generalised Brillouin zone itself is unchanged and only its parametrisation depends on $n$.

Projection methods are among the most widely used techniques for solving convex feasibility problems due to their simplicity and robustness. However, classical projection-based algorithms and existing circumcentered-reflection methods generally provide only asymptotic convergence. In this paper, we introduce a relaxed version of the centralized circumcentered-reflection method (cCRM) obtained through an extrapolation of the circumcenter step. We first analyze the two-set case and then extend the proposed approach to the successive framework for multiple convex sets, including under- and over-relaxation and a backtracking strategy. The convergence analysis establishes Fejér monotonicity and global convergence under suitable conditions, while local error-bound assumptions yield linear convergence and standard regularity assumptions imply superlinear convergence. Motivated by finitely convergent projection methods based on perturbed feasible sets, we further investigate a perturbed relaxed formulation designed to achieve finite termination. Numerical experiments show good practical performance, comparable to and competitive with the finitely convergent PACA method. The results also provide clear numerical evidence of finite termination in the tested instances.

Parameterized Quantum Circuits have emerged as a central framework for variational quantum algorithms in the NISQ regime, with major applications in quantum simulation, combinatorial optimization, and, more recently, quantum machine learning (QML). By developing a new approach for computing dynamical Lie algebras associated with quantum circuits combining parametrized gates and non-parametrized entangling gates, we aim to extend the applicability of recent algebraic results to architectures that are actually used in quantum machine learning.

We give an explicit example of a 9-dimensional 5-step nilpotent complex Lie algebra, generated by two elements, that is isomorphic to its complex conjugate but has no real form. In contrast to the earlier 10-dimensional examples, our proof is completely elementary and very short.

We construct a linear map from the space of spin polynomials in the critical Ising model on a lattice to the state space of the $M(4,3)$ minimal model of conformal field theory. The kernel of the map consists precisely of null fields, i.e., spin polynomials that vanish in all correlations. For a polynomial $\Op$ which is not a null field, we define its scaling dimension $\Delta_{\Op}\in\Z_{\geq0}\cup(\Z_{\geq0}+\frac{1}{8}),$ to be the lowest $\Loper{0}+\Loperbar{0}$ eigenvalue in the decomposition of its image into homogeneous components. We show that as the mesh size $\delta$ goes to zero, the rescaled field $\delta^{-\Delta_{\Op}}\Op$ has a non-trivial scaling limit, when inserted in correlations with other such fields. Moreover, all these limits can be expressed in terms of correlations of primary fields of $M(4,3)$ and the Virasoro action in the continuum. The highest weight vectors of $M(4,3)$, the joint eigenvectors of $\Loper{0}$ and $\Loperbar{0}$ with the same eigenvalues $0,1/16,1/2$ for both, are the images of the constant $1$, the spin at a single lattice vertex, and the energy at a single lattice edge, respectively. For the proof, we construct a map from a larger lattice field space, accommodating also fermion and disorder insertions, to a direct sum of two fermionic Fock spaces. This direct sum contains contains the state spaces of both the $M(4,3)$ and the free fermion CFT. The main technical ingredient of the proof is a new version of the exact fermionic current action on the space of lattice local fields due to C. Hongler, K.K. and F. Viklund, combined with some new advances in discrete complex analysis. Our results are valid on arbitrary isoradial graphs.

In this paper, by using signs rule for a class of special power series and for the Laplace integrals, we find the best power mean bounds for the hypergeometric function (sharp at the right endpoint). Our results contain many known conclusions.

Let $r_1,\ldots,r_t$ be positive integers, and let $\mathcal A$ be the set of integers of the form $2^{k_1^{r_1}}+\cdots+2^{k_t^{r_t}}$, where $k_1,\ldots,k_t$ are positive integers. For any positive integer $n$, let $r(n)$ be the number of representations $n=p+a$ with $p$ an odd prime and $a\in \mathcal A$. We prove that if $r_1^{-1}+\cdots+r_t^{-1}=1$, then, for every fixed positive integer $m$, the set of integers $n$ with $r(n)\ge m$ has positive lower asymptotic density. In particular, this confirms a conjecture of Chen.