We perform a systematic study of permutation statistics and bijective maps on permutations using SageMath to search the FindStat combinatorial statistics database to identify apparent instances of the cyclic sieving phenomenon (CSP). Cyclic sieving occurs on a set of objects, a statistic, and a map of order n n when the evaluation of the statistic generating function at the d d th power of the primitive n n th root of unity equals the number of fixed points under the d d th power of the map. Of the apparent instances found in our experiment, we prove 34 new instances of the CSP and conjecture three more. Our results are organized largely by orbit structure, proving instances of the CSP for involutions with 2 n − 1 2^{n-1} fixed points and 2 ⌊ n 2 ⌋ 2^{\lfloor \frac {n}{2}\rfloor } fixed points, as well as maps whose orbits all have the same size. The FindStat maps which exhibit the CSP include a map constructed by Corteel (using a bijection of Foata and Zeilberger) to swap the number of nestings and crossings, the invert Laguerre heap map, a map of Alexandersson and Kebede designed to preserve right-to-left minima, conjugation by the long cycle, as well as reverse, complement, rotation, Lehmer code rotation, and toric promotion. Our results combined with prior work of the last 5 authors in (Math. Comp. 93 , 921–976) show that, contrary to common expectations, actions that exhibit homomesy are not necessarily the best candidates for the CSP, and vice versa.

We study the algebraic complexity of Euclidean Distance (ED) minimization from a generic tensor to a variety of rank-one tensors. The ED degree of the Segre-Veronese variety counts the number of complex critical points of this optimization problem. We regard this invariant as a function of inner products. We prove that Frobenius inner product is a local minimum of the ED degree, and conjecture that it is a global minimum. We prove our conjecture in the case of matrices and symmetric binary and 3 × 3 × 3 3\times 3\times 3 tensors. We discuss the above optimization problem for other algebraic varieties, classifying all possible values of the ED degree. Our approach combines tools from Singularity Theory, Morse Theory, and Algebraic Geometry.

An isotropic vector of a given quadratic form is a nonzero vector where the form vanishes. Geometrically, it is a vector that is self-orthogonal with respect to this form. On the other hand, from the arithmetical point of view, an isotropic vector forms a solution to a multivariate quadratic equation. The problem of constructing isotropic vectors is one of the leading forces in the computational theory of quadratic forms. In this paper, we present algorithms for finding isotropic vectors of quadratic forms (of any dimension) over an arbitrary global field of characteristic distinct from 2 2 .

The 2 2 -variable polynomial ( y + 1 ) x 2 + ( y 2 + y + 1 ) x + ( y 2 + y ) (y + 1)x^2 + (y^2 + y + 1)x + (y^2 + y) has length 7 7 ; it is a factor of the length- 6 6 polynomial ( y + 1 ) x 3 + x 2 − y 3 x − y 3 − y 2 (y+1)x^3 + x^2 - y^3x - y^3 - y^2 which shares the same Mahler measure. On the other hand, consider the 2 2 -variable length- 7 7 polynomial ( 1 + x ) + ( 1 − x 2 + x 4 ) y + ( x 3 + x 4 ) y 2 (1+x) + (1-x^2+x^4)y + (x^3+x^4)y^2 . Extensive computations by Boyd and Mossinghoff [Experiment. Math. 14 (2005), pp. 403–414] suggested strongly that this has no length- 6 6 multiple with the same measure. But how can one prove this? In this paper we develop a method which attempts to find the shortest multiple of a polynomial in Z [ z 1 , … , z m ] \mathbb {Z}[z_1,\dots ,z_m] such that the multiple has the same Mahler measure as the original polynomial. The method is heuristic: it might fail (although we have yet to find an example when it does fail), but when it succeeds it provides a proof of shortness. In particular we can remove any doubt concerning the Boyd-Mossinghoff example mentioned above, and we are able to find shortest-possible Mahler-measure-preserving multiples of all the known 2 2 -dimensional examples having measure below 1.37.

We present new convergence analyses for parallel subspace correction methods for unconstrained semicoercive and nearly semicoercive convex optimization problems, generalizing the theory of singular and nearly singular linear problems to a class of nonlinear problems. Our results demonstrate that the elegant theoretical framework developed for singular and nearly singular linear problems can be extended to unconstrained semicoercive and nearly semicoercive convex optimization problems. For semicoercive problems, we show that the convergence rate can be estimated in terms of a seminorm stable decomposition over the subspaces and the kernel of the problem, aligning with the theory for singular linear problems. For nearly semicoercive problems, we establish a parameter-independent convergence rate, assuming the kernel of the semicoercive part can be decomposed into a sum of local kernels, which aligns with the theory for nearly singular problems. To demonstrate the applicability of our results, we provide convergence analyses of two-level additive Schwarz methods for solving certain nonlinear partial differential equations with Neumann boundary conditions, within the proposed abstract framework.

In this paper, we propose a novel adaptive stochastic extended iterative method, which can be viewed as an improved extension of the randomized extended Kaczmarz method, for finding the unique minimum Euclidean norm least-squares solution of a given linear system. In particular, we introduce three equivalent stochastic reformulations of the linear least-squares problem: stochastic unconstrained and constrained optimization problems, and the stochastic multiobjective optimization problem. We then alternately employ the adaptive variants of the stochastic heavy ball momentum (SHBM) method, which utilize iterative information to update the parameters, to solve the stochastic reformulations. We prove that our method converges R R -linearly in expectation, addressing an open problem in the literature related to designing theoretically supported adaptive SHBM methods. Numerical experiments show that our adaptive stochastic extended iterative method has strong advantages over the nonadaptive one.

By means of a Fourier optimization framework, we improve the current asymptotic bounds under the Generalized Riemann Hypothesis for two classical problems in number theory: the problem of estimating the least quadratic non-residue modulo a prime, and the problem of estimating the least prime in an arithmetic progression. In an appendix, we illustrate the connection between bounds for the argument of Dirichlet L L -functions on the critical line and bounds for the least quadratic non-residue.

This paper proposes a new multiple-scattering frequency-time hybrid (FTH-MS) integral equation solver for problems of wave scattering by obstacles in two dimensional space, including interior problems in closed cavities and problems exterior to a set of disconnected open or closed scattering obstacles. The multiple-scattering FTH-MS method is based on a partition of the domain boundary into a user-prescribed set of overlapping open arcs, along with a corresponding sequence of multiple-scattering problems that effectively decompose the interior problem into a series of open-arc wave equation subproblems. The new strategy provides a significant extension of the original FTH-MS algorithm originally presented by Bruno and Yin [Math. Comp. 93 (2024), pp. 551–587], in that (1) By allowing for use of an arbitrary of number of component arcs, and not just two as in the previous contribution, the new approach affords (1a) A significantly increased geometric flexibility, as well as (1b) The use of partitions for which each open arc leads to small numbers of iterations if iterative linear-algebra solvers are employed; and (2) It facilitates parallelization—as the subproblem solutions that are needed at each multiple scattering step can be evaluated in an embarrassingly parallel fashion. Utilizing a suitably-implemented Fourier transformation, each sub-problem is reduced to a Helmholtz frequency-domain problem that is tackled via a uniquely-solvable boundary integral equation. Similar FTH-MS methods are also presented for problems exterior to a number of bounded obstacles. All of the algorithms considered incorporate the previously introduced “time-windowing and recentering” methodology (that enables both treatment of incident signals of long duration and long time simulation), as well as a high-frequency Fourier transform algorithm that delivers numerically dispersionless, spectrally-accurate time evolution for arbitrarily long times.

We present a space-time finite element method for the heat equation that computes quasi-optimal approximations with respect to natural norms while incorporating local mesh refinements in space-time. The discretized problem is solved with a conjugate gradient method with an (nearly) optimal preconditioner.

We develop and analyze a splitting method for fluid-poroelastic structure interaction. The fluid is described using the Stokes equations and the poroelastic structure is described using the Biot equations. The transmission conditions on the interface are mass conservation, balance of stresses, and the Beavers-Joseph-Saffman condition. The splitting method involves single and decoupled Stokes and Biot solves at each time step. The sub-domain problems use Robin boundary conditions on the interface, which are obtained from the transmission conditions. The Robin data is represented by an auxiliary interface variable. We prove that the method is unconditionally stable and establish that the time discretization error is O ( T Δ t ) \mathcal {O}(\sqrt {T}\Delta t) , where T T is the final time and Δ t \Delta t is the time step. We further study the iterative version of the algorithm, which involves an iteration between the Stokes and Biot sub-problems at each time step. We prove that the iteration converges to a monolithic scheme with a Robin Lagrange multiplier used to impose the continuity of the velocity. Numerical experiments are presented to illustrate the theoretical results.