We revisit Ito’s [Osaka J. Math. 26 (1989), pp. 557–578] natural extension of the Farey tent map, which generates all regular continued fraction convergents and mediants of a given irrational. With a slight shift in perspective on the order in which these convergents and mediants arise, this natural extension is shown to provide an elegant and powerful tool in the metric theory of continued fractions. A wealth of old and new results—including limiting distributions of approximation coefficients, analogues of a theorem of Legendre and their refinements, and a generalisation of Lévy’s Theorem to subsequences of convergents and mediants—are presented as corollaries within this unifying theory.

We present an efficient and elementary algorithm for computing the number of primes up to N N in O ~ ( N ) \tilde {O}(\sqrt N) time, improving upon the existing combinatorial methods that require O ~ ( N 2 / 3 ) \tilde {O}(N ^{2/3}) time. Our method has a similar time complexity to the analytic approach to prime counting, while avoiding complex analysis and the use of arbitrary precision complex numbers. We apply our techniques to improve the state-of-the-art complexity of elementary algorithms for computing other number-theoretic functions, such as the Mertens function (in O ~ ( N ) \tilde {O}(\sqrt N) time compared to the known O ~ ( N 3 / 5 ) \tilde {O}(N^{3/5}) ), summing Euler’s totient function, counting square-free numbers and summing primes. Implementation code is provided.

We investigate a filtered Lie-Trotter splitting scheme for the “good” Boussinesq equation and derive an error estimate for initial data with very low regularity. Through the use of discrete Bourgain spaces, our analysis extends to initial data in H s H^{s} for 0 > s ≤ 2 0>s\leq 2 , overcoming the constraint of s > 1 / 2 s>1/2 imposed by the bilinear estimate in smooth Sobolev spaces. We establish convergence rates of order τ s / 2 \tau ^{s/2} in L 2 L^2 for such levels of regularity. Our analytical findings are supported by numerical experiments.

This paper introduces a new method for the efficient computation of oscillatory multidimensional lattice sums in geometries with boundaries. Such sums are ubiquitous in both pure and applied mathematics and have immediate applications in condensed matter and topological quantum physics. The challenge in their evaluation results from the combination of singular long-range interactions with the loss of translational invariance caused by the boundaries, rendering standard tools like Ewald summation ineffective. Our work shows that these lattice sums can be generated from a generalization of the Riemann zeta function to multidimensional non-periodic lattice sums. We put forth a new representation of this zeta function together with a numerical algorithm that ensures exponential convergence across an extensive range of geometries. Notably, our method’s runtime is influenced only by the complexity of the considered geometries and not by the number of particles, providing the foundation for efficient and precise simulations of macroscopic condensed matter systems. We showcase the practical utility of our method by computing interaction energies in a three-dimensional crystal structure with 3 × 10 23 3\times 10^{23} particles. Our method’s accuracy is demonstrated through extensive numerical experiments. A reference implementation is provided online along with this article.

We propose a unified theoretical framework to examine the energy dissipation properties at all stages of explicit exponential Runge-Kutta (EERK) methods for gradient flow problems. The main part of the novel framework is to construct the differential form of EERK method by using the difference coefficients of method and the so-called discrete orthogonal convolution kernels. As the main result, we prove that an EERK method can preserve the original energy dissipation law unconditionally if the associated differentiation matrix is positive semi-definite. A simple indicator, namely average dissipation rate, is also introduced for these multi-stage methods to evaluate the overall energy dissipation rate of an EERK method such that one can choose proper parameters in some parameterized EERK methods or compare different kinds of EERK methods. Some existing EERK methods in the literature are evaluated from the perspective of preserving the original energy dissipation law and the energy dissipation rate. Some numerical examples are also included to support our theory.

We compute a complete set of isomorphism classes of cubic fourfolds over F 2 \mathbb {F}_2 . Using this, we are able to compile statistics about various invariants of cubic fourfolds, including their counts of points, lines, and planes; all zeta functions of the smooth cubic fourfolds over F 2 \mathbb {F}_2 ; and their Newton polygons. One particular outcome is the number of smooth cubic fourfolds over F 2 \mathbb {F}_2 , which we fit into the asymptotic framework of discriminant complements. Another motivation is the realization problem for zeta functions of K 3 K3 surfaces. We present a refinement to the standard method of orbit enumeration that leverages filtrations and gives a significant speedup. In the case of cubic fourfolds, the relevant filtration is determined by Waring representation and the method brings the problem into the computationally tractable range.

The classical continuous mixed formulation of linear elasticity with pointwise symmetric stresses allows for a conforming finite element discretization with piecewise polynomials of degree at least three. Symmetric stress approximations of lower polynomial order are only possible when their div \operatorname {div} -conformity is weakened to the continuity of normal-normal components. In two dimensions, this condition is meant pointwise along edges for piecewise polynomials, but a corresponding characterization for general piecewise H ( div ) H(\operatorname {div}) tensors has been elusive. We introduce such a space and establish a continuous mixed formulation of linear planar elasticity with pointwise symmetric stresses that have, in a distributional sense, continuous normal-normal components across the edges of a shape-regular triangulation. The displacement is split into an L 2 L_2 field and a tangential trace on the skeleton of the mesh. The well-posedness of the new mixed formulation follows with a duality lemma relating the normal-normal continuous stresses with the tangential traces of displacements. For this new formulation we present a lowest-order conforming discretization. Stresses are approximated by piecewise quadratic symmetric tensors, whereas displacements are discretized by piecewise linear polynomials. The tangential displacement trace acts as a Lagrange multiplier and guarantees global div \operatorname {div} -conformity in the limit as the mesh-size tends to zero. We prove locking-free, quasi-optimal convergence of our scheme and illustrate this with numerical examples.

This paper focuses on multi-block optimization problems over transport polytopes, which underlie various applications including strongly correlated quantum physics and machine learning. Conventional block coordinate descent-type methods for the general multi-block problems store and operate on the matrix variables directly, resulting in formidable expenditure for large-scale settings. On the other hand, optimal transport problems, as a special case, have attracted extensive attention and numerical techniques that waive the use of the full matrices have recently emerged. However, it remains nontrivial to apply these techniques to the multi-block, possibly nonconvex problems with theoretical guarantees. In this work, we leverage the benefits of both sides and develop novel sampling-based block coordinate descent-type methods, which are equipped with either entropy regularization or Kullback-Leibler divergence. Each iteration of these methods solves subproblems restricted on the sampled degrees of freedom. Consequently, they involve only sparse matrices, which amounts to considerable complexity reductions. We explicitly characterize the sampling-induced errors and establish convergence and asymptotic properties for the methods equipped with the entropy regularization. Numerical experiments on typical strongly correlated electron systems corroborate their superior scalability over the methods utilizing full matrices. The advantage also enables the first visualization of approximate optimal transport maps between electron positions in three-dimensional contexts.

We prove a known 2-coloring of the integers [ N ] ≔ { 1 , 2 , 3 , … , N } [N] ≔\{1,2,3,\dots ,N\} minimizes the number of monochromatic arithmetic 3-progressions under certain restrictions. A monochromatic arithmetic progression is a set of equally-spaced integers that are all the same color. Previous work by Parrilo, Robertson and Saracino conjectured an optimal coloring for large N N that involves 12 12 colored blocks. Here, we prove that the conjectured coloring minimizes monochromatic arithmetic 3 3 -progressions among anti-symmetric colorings with 12 12 or fewer colored blocks. We leverage a connection to the coloring of the continuous interval [ 0 , 1 ] [0,1] used by Parrilo, Robertson, and Saracino as well as by Butler, Costello and Graham. Our proof identifies classes of colorings with permutations, then counts the permutations using mixed integer linear programming.

For the Gauss–Jacobi quadrature on [ − 1 , 1 ] [-1,1] , the location is estimated where the kernel of the error functional for functions analytic on an ellipse and its interior in the complex plane attains its maximum modulus. For the Jacobi weight ( 1 − t ) α ( 1 + t ) β (1-t)^\alpha (1+t)^\beta ( α > − 1 \alpha >-1 , β > − 1 \beta >-1 ) except for the Gegenbauer weight ( α = β \alpha =\beta ), the location is the intersection point of the ellipse with the real axis in the complex plane. For the Gegenbauer weight, it is the intersection point(s) with either the real or the imaginary axis or other axes with angle 1 4 π \tfrac {1}{4}\pi and 3 4 π \tfrac {3}{4}\pi . Our results support the empirical results provided by Gautschi and Varga [SIAM J. Numer. Anal, 20 (1983), pp. 1170–1186] for the Jacobi weight, the Gegenbauer weight and the Legendre weight. The results obtained are also illustrated numerically.