We study cut finite element discretizations of a Darcy interface problem based on the mixed finite element pairs $\textbf{RT}_k\times Q_k$, $k\geq 0$. Here $Q_k$ is the space of discontinuous polynomial functions of degree less or equal to $k$ and $\textbf{RT}$ is the Raviart-Thomas space. We show that the standard ghost penalty stabilization, often added in the weak forms of cut finite element methods for stability and control of the condition number of the linear system matrix, destroys the divergence-free property of the considered element pairs. Therefore, we propose new stabilization terms for the pressure and show that we recover the optimal approximation of the divergence without losing control of the condition number of the linear system matrix. We prove that with the new stabilization term the proposed cut finite element discretization results in pointwise divergence-free approximations of solenoidal velocity fields. We derive a priori error estimates for the proposed unfitted finite element discretization based on $\textbf{RT}_k\times Q_k$, $k\geq 0$. In addition, by decomposing the computational mesh into macro-elements and applying ghost penalty terms only on interior edges of macro-elements. Numerical experiments with element pairs $\textbf{RT}_0\times Q_0$, $\textbf{RT}_1\times Q_1$, and $\textbf{BDM}_1\times Q_0$. (where $\textbf{BDM}$ is the Brezzi-Douglas-Marini space) indicate that with the new method we have 1) optimal rates of convergence of the approximate velocity and pressure; 2) well-posed linear systems where the condition number of the system matrix scales as it does for fitted finite element discretizations; 3) optimal rates of convergence of the approximate divergence with pointwise divergence-free approximations of solenoidal velocity fields. All three properties hold independently of how the interface is positioned relative to the computational mesh.
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We introduce a novel formulation for the evolution of parametric curves by anisotropic curve shortening flow in ${\mathbb R}^d$, $d\geq2$. The reformulation hinges on a suitable manipulation of the parameterization's tangential velocity, leading to a strictly parabolic differential equation. Moreover, the derived equation is in divergence form, giving rise to a natural variational numerical method. For a fully discrete finite element approximation based on piecewise linear elements we prove optimal error estimates. Numerical simulations confirm the theoretical results and demonstrate the practicality of the method.
We give a lower bound of $\Omega(\sqrt n)$ on the unambiguous randomised parity-query complexity of the approximate majority problem -- that is, on the lowest randomised parity-query complexity of any function over $\{0,1\}^n$ whose value is "0" if the Hamming weight of the input is at most n/3, is "1" if the weight is at least 2n/3, and may be arbitrary otherwise.
We study propositional proof systems with inference rules that formalize restricted versions of the ability to make assumptions that hold without loss of generality, commonly used informally to shorten proofs. Each system we study is built on resolution. They are called BC${}^-$, RAT${}^-$, SBC${}^-$, and GER${}^-$, denoting respectively blocked clauses, resolution asymmetric tautologies, set-blocked clauses, and generalized extended resolution - all "without new variables." They may be viewed as weak versions of extended resolution (ER) since they are defined by first generalizing the extension rule and then taking away the ability to introduce new variables. Except for SBC${}^-$, they are known to be strictly between resolution and extended resolution. Several separations between these systems were proved earlier by exploiting the fact that they effectively simulate ER. We answer the questions left open: We prove exponential lower bounds for SBC${}^-$ proofs of a binary encoding of the pigeonhole principle, which separates ER from SBC${}^-$. Using this new separation, we prove that both RAT${}^-$ and GER${}^-$ are exponentially separated from SBC${}^-$. This completes the picture of their relative strengths.
In this article, we explore the spectral properties of general random kernel matrices $[K(U_i,U_j)]_{1\leq i\neq j\leq n}$ from a Lipschitz kernel $K$ with $n$ independent random variables $U_1,U_2,\ldots, U_n$ distributed uniformly over $[0,1]$. In particular we identify a dichotomy in the extreme eigenvalue of the kernel matrix, where, if the kernel $K$ is degenerate, the largest eigenvalue of the kernel matrix (after proper normalization) converges weakly to a weighted sum of independent chi-squared random variables. In contrast, for non-degenerate kernels, it converges to a normal distribution extending and reinforcing earlier results from Koltchinskii and Gin\'e (2000). Further, we apply this result to show a dichotomy in the asymptotic behavior of extreme eigenvalues of $W$-random graphs, which are pivotal in modeling complex networks and analyzing large-scale graph behavior. These graphs are generated using a kernel $W$, termed as graphon, by connecting vertices $i$ and $j$ with probability $W(U_i, U_j)$. Our results show that for a Lipschitz graphon $W$, if the degree function is constant, the fluctuation of the largest eigenvalue (after proper normalization) converges to the weighted sum of independent chi-squared random variables and an independent normal distribution. Otherwise, it converges to a normal distribution.
We prove that if $X,Y$ are positive, independent, non-Dirac random variables and if for $\alpha,\beta\ge 0$, $\alpha\neq \beta$, $$ \psi_{\alpha,\beta}(x,y)=\left(y\,\tfrac{1+\beta(x+y)}{1+\alpha x+\beta y},\;x\,\tfrac{1+\alpha(x+y)}{1+\alpha x+\beta y}\right), $$ then the random variables $U$ and $V$ defined by $(U,V)=\psi_{\alpha,\beta}(X,Y)$ are independent if and only if $X$ and $Y$ follow Kummer distributions with suitably related parameters. In other words, any invariant measure for a lattice recursion model governed by $\psi_{\alpha,\beta}$ in the scheme introduced by Croydon and Sasada in \cite{CS2020} is necessarily a product measure with Kummer marginals. The result extends earlier characterizations of Kummer and gamma laws by independence of $$ U=\tfrac{Y}{1+X}\quad\mbox{and}\quad V= X\left(1+\tfrac{Y}{1+X}\right), $$ which corresponds to the case of $\psi_{1,0}$. We also show that this independence property of Kummer laws covers, as limiting cases, several independence models known in the literature: the Lukacs, the Kummer-Gamma, the Matsumoto-Yor and the discrete Korteweg de Vries models.
The degree-diameter problem consists of finding the maximum number of vertices $n$ of a graph with diameter $d$ and maximum degree $\Delta$. This problem is well studied, and has been solved for plane graphs of low diameter in which every face is bounded by a 3-cycle (triangulations), and plane graphs in which every face is bounded by a 4-cycle (quadrangulations). In this paper, we solve the degree diameter problem for plane graphs of diameter 3 in which every face is bounded by a 5-cycle (pentagulations). We prove that if $\Delta \geq 8$, then $n \leq 3\Delta - 1$ for such graphs. This bound is sharp for $\Delta$ odd.
Among semiparametric regression models, partially linear additive models provide a useful tool to include additive nonparametric components as well as a parametric component, when explaining the relationship between the response and a set of explanatory variables. This paper concerns such models under sparsity assumptions for the covariates included in the linear component. Sparse covariates are frequent in regression problems where the task of variable selection is usually of interest. As in other settings, outliers either in the residuals or in the covariates involved in the linear component have a harmful effect. To simultaneously achieve model selection for the parametric component of the model and resistance to outliers, we combine preliminary robust estimators of the additive component, robust linear $MM-$regression estimators with a penalty such as SCAD on the coefficients in the parametric part. Under mild assumptions, consistency results and rates of convergence for the proposed estimators are derived. A Monte Carlo study is carried out to compare, under different models and contamination schemes, the performance of the robust proposal with its classical counterpart. The obtained results show the advantage of using the robust approach. Through the analysis of a real data set, we also illustrate the benefits of the proposed procedure.
Suitable discretizations through tensor product formulas of popular multidimensional operators (diffusion or diffusion--advection, for instance) lead to matrices with $d$-dimensional Kronecker sum structure. For evolutionary Partial Differential Equations containing such operators and integrated in time with exponential integrators, it is then of paramount importance to efficiently approximate the actions of $\varphi$-functions of the arising matrices. In this work, we show how to produce directional split approximations of third order with respect to the time step size. They conveniently employ tensor-matrix products (the so-called $\mu$-mode product and related Tucker operator, realized in practice with high performance level 3 BLAS), and allow for the effective usage of exponential Runge--Kutta integrators up to order three. The technique can also be efficiently implemented on modern computer hardware such as Graphic Processing Units. The approach has been successfully tested against state-of-the-art techniques on two well-known physical models that lead to Turing patterns, namely the 2D Schnakenberg and the 3D FitzHugh--Nagumo systems, on different architectures.
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.
The Gaussian chain graph model simultaneously parametrizes (i) the direct effects of $p$ predictors on $q$ outcomes and (ii) the residual partial covariances between pairs of outcomes. We introduce a new method for fitting sparse Gaussian chain graph models with spike-and-slab LASSO (SSL) priors. We develop an Expectation Conditional Maximization algorithm to obtain sparse estimates of the $p \times q$ matrix of direct effects and the $q \times q$ residual precision matrix. Our algorithm iteratively solves a sequence of penalized maximum likelihood problems with self-adaptive penalties that gradually filter out negligible regression coefficients and partial covariances. Because it adaptively penalizes individual model parameters, our method is seen to outperform fixed-penalty competitors on simulated data. We establish the posterior contraction rate for our model, buttressing our method's excellent empirical performance with strong theoretical guarantees. Using our method, we estimated the direct effects of diet and residence type on the composition of the gut microbiome of elderly adults.
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