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In this paper, we formulate and analyse a geometric low-regularity integrator for solving the nonlinear Klein-Gordon equation in the $d$-dimensional space with $d=1,2,3$. The integrator is constructed based on the two-step trigonometric method and thus it has a simple form. Error estimates are rigorously presented to show that the integrator can achieve second-order time accuracy in the energy space under the regularity requirement in $H^{1+\frac{d}{4}}\times H^{\frac{d}{4}}$. Moreover, the time symmetry of the scheme ensures its good long-time energy conservation which is rigorously proved by the technique of modulated Fourier expansions. A numerical test is presented and the numerical results demonstrate the superiorities of the new integrator over some existing methods.

In this paper, we consider a numerical method for the multi-term Caputo-Fabrizio time-fractional diffusion equations (with orders $\alpha_i\in(0,1)$, $i=1,2,\cdots,n$). The proposed method employs a fast finite difference scheme to approximate multi-term fractional derivatives in time, requiring only $O(1)$ storage and $O(N_T)$ computational complexity, where $N_T$ denotes the total number of time steps. Then we use a Legendre spectral collocation method for spatial discretization. The stability and convergence of the scheme have been thoroughly discussed and rigorously established. We demonstrate that the proposed scheme is unconditionally stable and convergent with an order of $O(\left(\Delta t\right)^{2}+N^{-m})$, where $\Delta t$, $N$, and $m$ represent the timestep size, polynomial degree, and regularity in the spatial variable of the exact solution, respectively. Numerical results are presented to validate the theoretical predictions.

Statistical inverse learning aims at recovering an unknown function $f$ from randomly scattered and possibly noisy point evaluations of another function $g$, connected to $f$ via an ill-posed mathematical model. In this paper we blend statistical inverse learning theory with the classical regularization strategy of applying finite-dimensional projections. Our key finding is that coupling the number of random point evaluations with the choice of projection dimension, one can derive probabilistic convergence rates for the reconstruction error of the maximum likelihood (ML) estimator. Convergence rates in expectation are derived with a ML estimator complemented with a norm-based cut-off operation. Moreover, we prove that the obtained rates are minimax optimal.

We introduce a single-set axiomatisation of cubical $\omega$-categories, including connections and inverses. We justify these axioms by establishing a series of equivalences between the category of single-set cubical $\omega$-categories, and their variants with connections and inverses, and the corresponding cubical $\omega$-categories. We also report on the formalisation of cubical $\omega$-categories with the Isabelle/HOL proof assistant, which has been instrumental in finding the single-set axioms.

We study the algorithmic undecidability of abstract dynamical properties for sofic $\mathbb{Z}^{2}$-subshifts and subshifts of finite type (SFTs) on $\mathbb{Z}^{2}$. Within the class of sofic $\mathbb{Z}^{2}$-subshifts, we prove the undecidability of every nontrivial dynamical property. We show that although this is not the case for $\mathbb{Z}^{2}$-SFTs, it is still possible to establish the undecidability of a large class of dynamical properties. This result is analogous to the Adian-Rabin undecidability theorem for group properties. Besides dynamical properties, we consider dynamical invariants of $\mathbb{Z}^{2}$-SFTs taking values in partially ordered sets. It is well known that the topological entropy of a $\mathbb{Z}^{2}$-SFT can not be effectively computed from an SFT presentation. We prove a generalization of this result to \emph{every} dynamical invariant which is nonincreasing by factor maps, and satisfies a mild additional technical condition. Our results are also valid for $\Z^{d}$, $d\geq2$, and more generally for any group where determining whether a subshift of finite type is empty is undecidable.

We generalize the Poisson limit theorem to binary functions of random objects whose law is invariant under the action of an amenable group. Examples include stationary random fields, exchangeable sequences, and exchangeable graphs. A celebrated result of E. Lindenstrauss shows that normalized sums over certain increasing subsets of such groups approximate expectations. Our results clarify that the corresponding unnormalized sums of binary statistics are asymptotically Poisson, provided suitable mixing conditions hold. They extend further to randomly subsampled sums and also show that strict invariance of the distribution is not needed if the requisite mixing condition defined by the group holds. We illustrate the results with applications to random fields, Cayley graphs, and Poisson processes on groups.

This paper studies the convergence of a spatial semidiscretization of a three-dimensional stochastic Allen-Cahn equation with multiplicative noise. For non-smooth initial values, the regularity of the mild solution is investigated, and an error estimate is derived with the spatial $ L^2 $-norm. For smooth initial values, two error estimates with the general spatial $ L^q $-norms are established.

SARRIGUREN, a new complete algorithm for SAT based on counting clauses (which is valid also for Unique-SAT and #SAT) is described, analyzed and tested. Although existing complete algorithms for SAT perform slower with clauses with many literals, that is an advantage for SARRIGUREN, because the more literals are in the clauses the bigger is the probability of overlapping among clauses, a property that makes the clause counting process more efficient. Actually, it provides a $O(m^2 \times n/k)$ time complexity for random $k$-SAT instances of $n$ variables and $m$ relatively dense clauses, where that density level is relative to the number of variables $n$, that is, clauses are relatively dense when $k\geq7\sqrt{n}$. Although theoretically there could be worst-cases with exponential complexity, the probability of those cases to happen in random $k$-SAT with relatively dense clauses is practically zero. The algorithm has been empirically tested and that polynomial time complexity maintains also for $k$-SAT instances with less dense clauses ($k\geq5\sqrt{n}$). That density could, for example, be of only 0.049 working with $n=20000$ variables and $k=989$ literals. In addition, they are presented two more complementary algorithms that provide the solutions to $k$-SAT instances and valuable information about number of solutions for each literal. Although this algorithm does not solve the NP=P problem (it is not a polynomial algorithm for 3-SAT), it broads the knowledge about that subject, because $k$-SAT with $k>3$ and dense clauses is not harder than 3-SAT. Moreover, the Python implementation of the algorithms, and all the input datasets and obtained results in the experiments are made available.

High-frequency issues have been remarkably challenges in numerical methods for partial differential equations. In this paper, a learning based numerical method (LbNM) is proposed for Helmholtz equation with high frequency. The main novelty is using Tikhonov regularization method to stably learn the solution operator by utilizing relevant information especially the fundamental solutions. Then applying the solution operator to a new boundary input could quickly update the solution. Based on the method of fundamental solutions and the quantitative Runge approximation, we give the error estimate. This indicates interpretability and generalizability of the present method. Numerical results validates the error analysis and demonstrates the high-precision and high-efficiency features.

We present approximation algorithms for the Fault-tolerant $k$-Supplier with Outliers ($\mathsf{F}k\mathsf{SO}$) problem. This is a common generalization of two known problems -- $k$-Supplier with Outliers, and Fault-tolerant $k$-Supplier -- each of which generalize the well-known $k$-Supplier problem. In the $k$-Supplier problem the goal is to serve $n$ clients $C$, by opening $k$ facilities from a set of possible facilities $F$; the objective function is the farthest that any client must travel to access an open facility. In $\mathsf{F}k\mathsf{SO}$, each client $v$ has a fault-tolerance $\ell_v$, and now desires $\ell_v$ facilities to serve it; so each client $v$'s contribution to the objective function is now its distance to the $\ell_v^{\text{th}}$ closest open facility. Furthermore, we are allowed to choose $m$ clients that we will serve, and only those clients contribute to the objective function, while the remaining $n-m$ are considered outliers. Our main result is a $\min\{4t-1,2^t+1\}$-approximation for the $\mathsf{F}k\mathsf{SO}$ problem, where $t$ is the number of distinct values of $\ell_v$ that appear in the instance. At $t=1$, i.e. in the case where the $\ell_v$'s are uniformly some $\ell$, this yields a $3$-approximation, improving upon the $11$-approximation given for the uniform case by Inamdar and Varadarajan [2020], who also introduced the problem. Our result for the uniform case matches tight $3$-approximations that exist for $k$-Supplier, $k$-Supplier with Outliers, and Fault-tolerant $k$-Supplier. Our key technical contribution is an application of the round-or-cut schema to $\mathsf{F}k\mathsf{SO}$. Guided by an LP relaxation, we reduce to a simpler optimization problem, which we can solve to obtain distance bounds for the "round" step, and valid inequalities for the "cut" step.