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This paper introduces a formulation of the variable density incompressible Navier-Stokes equations by modifying the nonlinear terms in a consistent way. For Galerkin discretizations, the formulation leads to full discrete conservation of mass, squared density, momentum, angular momentum and kinetic energy without the divergence-free constraint being strongly enforced. In addition to favorable conservation properties, the formulation is shown to make the density field invariant to global shifts. The effect of viscous regularizations on conservation properties is also investigated. Numerical tests validate the theory developed in this work. The new formulation shows superior performance compared to other formulations from the literature, both in terms of accuracy for smooth problems and in terms of robustness.

This paper deals with the problem of numerically computing the roots of polynomials $p_k(x)$, $k=1,2,\ldots$, of degree $n=2^k-1$ recursively defined by $p_1(x)=x+1$, $p_k(x)=xp_{k-1}(x)^2+1$. An algorithm based on the Ehrlich-Aberth simultaneous iterations complemented by the Fast Multi-pole Method and the fast search of near neighbors of a set of complex numbers is provided. The algorithm, which relies on a specific strategy of selecting initial approximations, costs $O(n\log n)$ arithmetic operations per step. A Fortran 95 implementation is given and numerical experiments are carried out. Experimentally, it turns out that the number of iterations needed to arrive at numerical convergence is $O(\log n)$. This allows us to compute the roots of $p_k(x)$ up to degree $n=2^{24}-1$ in about 16 minutes on a laptop with 16 GB RAM, and up to degree $n=2^{28}-1$ in about 69 minutes on a machine with 256 GB RAM. The case of degree $n=2^{30}-1$ would require higher memory and higher precision to separate the roots. With a suitable adaptation of FMM to the limit of 256 GB RAM and by performing the computation in extended precision (i.e. with 10-byte floating point representation) we were able to compute all the roots in about two weeks of CPU time for $n=2^{30}-1$. From the experimental analysis, explicit asymptotic expressions of the real roots of $p_k(x)$ and an explicit expression of $\min_{i\ne j}|\xi_i^{(k)}-\xi_j^{(k)}|$ for the roots $\xi_i^{(k)}$ of $p_k(x)$ are deduced. The approach is effectively applied to general classes of polynomials defined by a doubling recurrence.

We present difference schemes for stochastic transport equations with low-regularity velocity fields. We establish $L^2$ stability and convergence of the difference approximations under conditions that are less strict than those required for deterministic transport equations. The $L^2$ estimate, crucial for the analysis, is obtained through a discrete duality argument and a comprehensive examination of a class of backward parabolic difference schemes.

This short note shows the superconvergence of an $H(\mathrm{grad}\,\mathrm{curl})$-nonconforming brick element very recently introduced in [17] for the quad-curl problem. The supercloseness is based on proper modifications for both the interpolation and the discrete formulation, leading to an $O(h^2)$ superclose order in the discrete $H(\mathrm{grad}\,\mathrm{curl})$ norm. Moreover, we propose a suitable postprocessing method to ensure the global superconvergence. Numerical results verify our theory.

Challenges with data in the big-data era include (i) the dimension $p$ is often larger than the sample size $n$ (ii) outliers or contaminated points are frequently hidden and more difficult to detect. Challenge (i) renders most conventional methods inapplicable. Thus, it attracts tremendous attention from statistics, computer science, and bio-medical communities. Numerous penalized regression methods have been introduced as modern methods for analyzing high-dimensional data. Disproportionate attention has been paid to the challenge (ii) though. Penalized regression methods can do their job very well and are expected to handle the challenge (ii) simultaneously. Most of them, however, can break down by a single outlier (or single adversary contaminated point) as revealed in this article. The latter systematically examines leading penalized regression methods in the literature in terms of their robustness, provides quantitative assessment, and reveals that most of them can break down by a single outlier. Consequently, a novel robust penalized regression method based on the least sum of squares of depth trimmed residuals is proposed and studied carefully. Experiments with simulated and real data reveal that the newly proposed method can outperform some leading competitors in estimation and prediction accuracy in the cases considered.

A $hole$ is an induced cycle of length at least four, and an odd hole is a hole of odd length. A {\em fork} is a graph obtained from $K_{1,3}$ by subdividing an edge once. An {\em odd balloon} is a graph obtained from an odd hole by identifying respectively two consecutive vertices with two leaves of $K_{1, 3}$. A {\em gem} is a graph that consists of a $P_4$ plus a vertex adjacent to all vertices of the $P_4$. A {\em butterfly} is a graph obtained from two traingles by sharing exactly one vertex. A graph $G$ is perfectly divisible if for each induced subgraph $H$ of $G$, $V(H)$ can be partitioned into $A$ and $B$ such that $H[A]$ is perfect and $\omega(H[B])<\omega(H)$. In this paper, we show that (odd balloon, fork)-free graphs are perfectly divisible (this generalizes some results of Karthick {\em et al}). As an application, we show that $\chi(G)\le\binom{\omega(G)+1}{2}$ if $G$ is (fork, gem)-free or (fork, butterfly)-free.

We consider the minimal thermodynamic cost of an individual computation, where a single input $x$ is mapped to a single output $y$. In prior work, Zurek proposed that this cost was given by $K(x\vert y)$, the conditional Kolmogorov complexity of $x$ given $y$ (up to an additive constant which does not depend on $x$ or $y$). However, this result was derived from an informal argument, applied only to deterministic computations, and had an arbitrary dependence on the choice of protocol (via the additive constant). Here we use stochastic thermodynamics to derive a generalized version of Zurek's bound from a rigorous Hamiltonian formulation. Our bound applies to all quantum and classical processes, whether noisy or deterministic, and it explicitly captures the dependence on the protocol. We show that $K(x\vert y)$ is a minimal cost of mapping $x$ to $y$ that must be paid using some combination of heat, noise, and protocol complexity, implying a tradeoff between these three resources. Our result is a kind of "algorithmic fluctuation theorem" with implications for the relationship between the Second Law and the Physical Church-Turing thesis.

We consider problems of minimizing functionals $\mathcal{F}$ of probability measures on the Euclidean space. To propose an accelerated gradient descent algorithm for such problems, we consider gradient flow of transport maps that give push-forward measures of an initial measure. Then we propose a deterministic accelerated algorithm by extending Nesterov's acceleration technique with momentum. This algorithm do not based on the Wasserstein geometry. Furthermore, to estimate the convergence rate of the accelerated algorithm, we introduce new convexity and smoothness for $\mathcal{F}$ based on transport maps. As a result, we can show that the accelerated algorithm converges faster than a normal gradient descent algorithm. Numerical experiments support this theoretical result.

At STOC 2002, Eiter, Gottlob, and Makino presented a technique called ordered generation that yields an $n^{O(d)}$-delay algorithm listing all minimal transversals of an $n$-vertex hypergraph of degeneracy $d$. Recently at IWOCA 2019, Conte, Kant\'e, Marino, and Uno asked whether this XP-delay algorithm parameterized by $d$ could be made FPT-delay parameterized by $d$ and the maximum degree $\Delta$, i.e., an algorithm with delay $f(d,\Delta)\cdot n^{O(1)}$ for some computable function $f$. Moreover, as a first step toward answering that question, they note that the same delay is open for the intimately related problem of listing all minimal dominating sets in graphs. In this paper, we answer the latter question in the affirmative.

A standard approach to solve ordinary differential equations, when they describe dynamical systems, is to adopt a Runge-Kutta or related scheme. Such schemes, however, are not applicable to the large class of equations which do not constitute dynamical systems. In several physical systems, we encounter integro-differential equations with memory terms where the time derivative of a state variable at a given time depends on all past states of the system. Secondly, there are equations whose solutions do not have well-defined Taylor series expansion. The Maxey-Riley-Gatignol equation, which describes the dynamics of an inertial particle in nonuniform and unsteady flow, displays both challenges. We use it as a test bed to address the questions we raise, but our method may be applied to all equations of this class. We show that the Maxey-Riley-Gatignol equation can be embedded into an extended Markovian system which is constructed by introducing a new dynamical co-evolving state variable that encodes memory of past states. We develop a Runge-Kutta algorithm for the resultant Markovian system. The form of the kernels involved in deriving the Runge-Kutta scheme necessitates the use of an expansion in powers of $t^{1/2}$. Our approach naturally inherits the benefits of standard time-integrators, namely a constant memory storage cost, a linear growth of operational effort with simulation time, and the ability to restart a simulation with the final state as the new initial condition.