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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.

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.

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.

In the $0$-Extension problem, we are given an edge-weighted graph $G=(V,E,c)$, a set $T\subseteq V$ of its vertices called terminals, and a semi-metric $D$ over $T$, and the goal is to find an assignment $f$ of each non-terminal vertex to a terminal, minimizing the sum, over all edges $(u,v)\in E$, the product of the edge weight $c(u,v)$ and the distance $D(f(u),f(v))$ between the terminals that $u,v$ are mapped to. Current best approximation algorithms on $0$-Extension are based on rounding a linear programming relaxation called the \emph{semi-metric LP relaxation}. The integrality gap of this LP, with best upper bound $O(\log |T|/\log\log |T|)$ and best lower bound $\Omega((\log |T|)^{2/3})$, has been shown to be closely related to the best quality of cut and flow vertex sparsifiers. We study a variant of the $0$-Extension problem where Steiner vertices are allowed. Specifically, we focus on the integrality gap of the same semi-metric LP relaxation to this new problem. Following from previous work, this new integrality gap turns out to be closely related to the quality achievable by cut/flow vertex sparsifiers with Steiner nodes, a major open problem in graph compression. Our main result is that the new integrality gap stays superconstant $\Omega(\log\log |T|)$ even if we allow a super-linear $O(|T|\log^{1-\varepsilon}|T|)$ number of Steiner nodes.

We introduce a strict saddle property for $\ell_p$ regularized functions, and propose an iterative reweighted $\ell_1$ algorithm to solve the $\ell_p$ regularized problems. The algorithm is guaranteed to converge only to local minimizers when randomly initialized. The strict saddle property is shown generic on these sparse optimization problems. Those analyses as well as the proposed algorithm can be easily extended to general nonconvex regularized problems.

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.

Dirac delta distributionally sourced differential equations emerge in many dynamical physical systems from neuroscience to black hole perturbation theory. Most of these lack exact analytical solutions and are thus best tackled numerically. This work describes a generic numerical algorithm which constructs discontinuous spatial and temporal discretisations by operating on discontinuous Lagrange and Hermite interpolation formulae recovering higher order accuracy. It is shown by solving the distributionally sourced wave equation, which has analytical solutions, that numerical weak-form solutions can be recovered to high order accuracy by solving a first-order reduced system of ordinary differential equations. The method-of-lines framework is applied to the DiscoTEX algorithm i.e through discontinuous collocation with implicit-turned-explicit (IMTEX) integration methods which are symmetric and conserve symplectic structure. Furthermore, the main application of the algorithm is proved, for the first-time, by calculating the amplitude at any desired location within the numerical grid, including at the position (and at its right and left limit) where the wave- (or wave-like) equation is discontinuous via interpolation using DiscoTEX. This is shown, firstly by solving the wave- (or wave-like) equation and comparing the numerical weak-form solution to the exact solution. Finally, one shows how to reconstruct the scalar and gravitational metric perturbations from weak-form numerical solutions of a non-rotating black hole, which do not have known exact analytical solutions, and compare against state-of-the-art frequency domain results. One concludes by motivating how DiscoTEX, and related algorithms, open a promising new alternative Extreme-Mass-Ratio-Inspiral (EMRI)s waveform generation route via a self-consistent evolution for the gravitational self-force programme in the time-domain.