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In this study, we consider a class of linear matroid interdiction problems, where the feasible sets for the upper-level decision-maker (referred to as the leader) and the lower-level decision-maker (referred to as the follower) are given by partition matroids with a common ground set. In contrast to classical network interdiction models where the leader is subject to a single budget constraint, in our setting, both the leader and the follower are subject to several independent cardinality constraints and engage in a zero-sum game. While a single-level linear integer programming problem over a partition matroid is known to be polynomially solvable, we prove that the considered bilevel problem is NP-hard, even when the objective function coefficients are all binary. On a positive note, it turns out that, if the number of cardinality constraints is fixed for either the leader or the follower, then the considered class of bilevel problems admits several polynomial-time solution schemes. Specifically, these schemes are based on a single-level dual reformulation, a dynamic programming-based approach, and a 2-flip local search algorithm for the leader.

Recently, a family of unconventional integrators for ODEs with polynomial vector fields was proposed, based on the polarization of vector fields. The simplest instance is the by now famous Kahan discretization for quadratic vector fields. All these integrators seem to possess remarkable conservation properties. In particular, it has been proved that, when the underlying ODE is Hamiltonian, its polarization discretization possesses an integral of motion and an invariant volume form. In this note, we propose a new algebraic approach to derivation of the integrals of motion for polarization discretizations.

We study the existence and uniqueness of Lp-bounded mild solutions for a class ofsemilinear stochastic evolutions equations driven by a real L\'evy processes withoutGaussian component not square integrable for instance the stable process through atruncation method by separating the big and small jumps together with the classicaland simple Banach fixed point theorem ; under local Lipschitz, Holder, linear growthconditions on the coefficients.

Comparisons of frequency distributions often invoke the concept of shift to describe directional changes in properties such as the mean. In the present study, we sought to define shift as a property in and of itself. Specifically, we define distributional shift (DS) as the concentration of frequencies away from the discrete class having the greatest value (e.g., the right-most bin of a histogram). We derive a measure of DS using the normalized sum of exponentiated cumulative frequencies. We then define relative distributional shift (RDS) as the difference in DS between two distributions, revealing the magnitude and direction by which one distribution is concentrated to lesser or greater discrete classes relative to another. We find that RDS is highly related to popular measures that, while based on the comparison of frequency distributions, do not explicitly consider shift. While RDS provides a useful complement to other comparative measures, DS allows shift to be quantified as a property of individual distributions, similar in concept to a statistical moment.

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 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.

We study discretizations of fractional fully nonlinear equations by powers of discrete Laplacians. Our problems are parabolic and of order $\sigma\in(0,2)$ since they involve fractional Laplace operators $(-\Delta)^{\sigma/2}$. They arise e.g.~in control and game theory as dynamic programming equations, and solutions are non-smooth in general and should be interpreted as viscosity solutions. Our approximations are realized as finite-difference quadrature approximations and are 2nd order accurate for all values of $\sigma$. The accuracy of previous approximations depend on $\sigma$ and are worse when $\sigma$ is close to $2$. We show that the schemes are monotone, consistent, $L^\infty$-stable, and convergent using a priori estimates, viscosity solutions theory, and the method of half-relaxed limits. We present several numerical examples.

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.

We present a novel discontinuous Galerkin finite element method for numerical simulations of the rotating thermal shallow water equations in complex geometries using curvilinear meshes, with arbitrary accuracy. We derive an entropy functional which is convex, and which must be preserved in order to preserve model stability at the discrete level. The numerical method is provably entropy stable and conserves mass, buoyancy, vorticity, and energy. This is achieved by using novel entropy stable numerical fluxes, summation-by-parts principle, and splitting the pressure and convection operators so that we can circumvent the use of chain rule at the discrete level. Numerical simulations on a cubed sphere mesh are presented to verify the theoretical results. The numerical experiments demonstrate the robustness of the method for a regime of well developed turbulence, where it can be run stably without any dissipation. The entropy stable fluxes are sufficient to control the grid scale noise generated by geostrophic turbulence, eliminating the need for artificial stabilisation.

We develop a novel and efficient discontinuous Galerkin spectral element method (DG-SEM) for the spherical rotating shallow water equations in vector invariant form. We prove that the DG-SEM is energy stable, and discretely conserves mass, vorticity, and linear geostrophic balance on general curvlinear meshes. These theoretical results are possible due to our novel entropy stable numerical DG fluxes for the shallow water equations in vector invariant form. We experimentally verify these results on a cubed sphere mesh. Additionally, we show that our method is robust, that is can be run stably without any dissipation. The entropy stable fluxes are sufficient to control the grid scale noise generated by geostrophic turbulence without the need for artificial stabilisation.