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In this paper we construct high order numerical methods for solving third and fourth orders nonlinear functional differential equations (FDE). They are based on the discretization of iterative methods on continuous level with the use of the trapezoidal quadrature formulas with corrections. Depending on the number of terms in the corrections we obtain methods of $O(h^4)$ and $O(h^6)$ accuracy. Some numerical experiments demonstrate the validity of the obtained theoretical results. The approach used here for the third and fourth orders nonlinear functional differential equations can be applied to functional differential equations of any orders.

We propose a tamed-adaptive Milstein scheme for stochastic differential equations in which the first-order derivatives of the coefficients are locally H\"older continuous of order $\alpha$. We show that the scheme converges in the $L_2$-norm with a rate of $(1+\alpha)/2$ over both finite intervals $[0, T]$ and the infinite interval $(0, +\infty)$, under certain growth conditions on the coefficients.

We extend prior work comparing linear multilevel models (MLM) and fixed effect (FE) models to the generalized linear model (GLM) setting, where the coefficient on a treatment variable is of primary interest. This leads to three key insights. (i) First, as in the linear setting, MLM can be thought of as a regularized form of FE. This explains why MLM can show large biases in its treatment coefficient estimates when group-level confounding is present. However, unlike the linear setting, there is not an exact equivalence between MLM and regularized FE coefficient estimates in GLMs. (ii) Second, we study a generalization of "bias-corrected MLM" (bcMLM) to the GLM setting. Neither FE nor bcMLM entirely solves MLM's bias problem in GLMs, but bcMLM tends to show less bias than does FE. (iii) Third, and finally, just like in the linear setting, MLM's default standard errors can misspecify the true intragroup dependence structure in the GLM setting, which can lead to downwardly biased standard errors. A cluster bootstrap is a more agnostic alternative. Ultimately, for non-linear GLMs, we recommend bcMLM for estimating the treatment coefficient, and a cluster bootstrap for standard errors and confidence intervals. If a bootstrap is not computationally feasible, then we recommend FE with cluster-robust standard errors.

In this paper we consider a class of conjugate discrete-time Riccati equations (CDARE), arising originally from the linear quadratic regulation problem for discrete-time antilinear systems. Recently, we have proved the existence of the maximal solution to the CDARE with a nonsingular control weighting matrix under the framework of the constructive method. Our contribution in the work is to finding another meaningful Hermitian solutions, which has received little attention in this topic. Moreover, we show that some extremal solutions cannot be attained at the same time, and almost (anti-)stabilizing solutions coincide with some extremal solutions. It is to be expected that our theoretical results presented in this paper will play an important role in the optimal control problems for discrete-time antilinear systems.

We present and analyze two stabilized finite element methods for solving numerically the Poisson--Nernst--Planck equations. The stabilization we consider is carried out by using a shock detector and a discrete graph Laplacian operator for the ion equations, whereas the discrete equation for the electric potential need not be stabilized. Discrete solutions stemmed from the first algorithm preserve both maximum and minimum discrete principles. For the second algorithm, its discrete solutions are conceived so that they hold discrete principles and obey an entropy law provided that an acuteness condition is imposed for meshes. Remarkably the latter is found to be unconditionally stable. We validate our methodology through numerical experiments.

In this note, when the dimension $p$ is large we look into the insight of the Mar$\check{c}$enko-Pastur equation to get an explicit equality relationship, and use the obtained equality to establish a new kind of orthogonally equivariant estimator of the population covariance matrix. Under some regularity conditions, the proposed novel estimators of the population eigenvalues are shown to be consistent for the eigenvalues of population covariance matrix. It is also shown that the proposed estimator is the best orthogonally equivariant estimator of population covariance matrix under the normalized Stein loss function.

QAC$^0$ is the class of constant-depth quantum circuits with polynomially many ancillary qubits, where Toffoli gates on arbitrarily many qubits are allowed. In this work, we show that the parity function cannot be computed in QAC$^0$, resolving a long-standing open problem in quantum circuit complexity more than twenty years old. As a result, this proves ${\rm QAC}^0 \subsetneqq {\rm QAC}_{\rm wf}^0$. We also show that any QAC circuit of depth $d$ that approximately computes parity on $n$ bits requires $2^{\widetilde{\Omega}(n^{1/d})}$ ancillary qubits, which is close to tight. This implies a similar lower bound on approximately preparing cat states using QAC circuits. Finally, we prove a quantum analog of the Linial-Mansour-Nisan theorem for QAC$^0$. This implies that, for any QAC$^0$ circuit $U$ with $a={\rm poly}(n)$ ancillary qubits, and for any $x\in\{0,1\}^n$, the correlation between $Q(x)$ and the parity function is bounded by ${1}/{2} + 2^{-\widetilde{\Omega}(n^{1/d})}$, where $Q(x)$ denotes the output of measuring the output qubit of $U|x,0^a\rangle$. All the above consequences rely on the following technical result. If $U$ is a QAC$^0$ circuit with $a={\rm poly}(n)$ ancillary qubits, then there is a distribution $\mathcal{D}$ of bounded polynomials of degree polylog$(n)$ such that with high probability, a random polynomial from $\mathcal{D}$ approximates the function $\langle x,0^a| U^\dag Z_{n+1} U |x,0^a\rangle$ for a large fraction of $x\in \{0,1\}^n$. This result is analogous to the Razborov-Smolensky result on the approximation of AC$^0$ circuits by random low-degree polynomials.

We consider the problems of testing and learning an unknown $n$-qubit Hamiltonian $H$ from queries to its evolution operator $e^{-iHt}$ under the normalized Frobenius norm. We prove: 1. Local Hamiltonians: We give a tolerant testing protocol to decide if $H$ is $\epsilon_1$-close to $k$-local or $\epsilon_2$-far from $k$-local, with $O(1/(\epsilon_2-\epsilon_1)^{4})$ queries, solving open questions posed in a recent work by Bluhm et al. For learning a $k$-local $H$ up to error $\epsilon$, we give a protocol with query complexity $\exp(O(k^2+k\log(1/\epsilon)))$ independent of $n$, by leveraging the non-commutative Bohnenblust-Hille inequality. 2. Sparse Hamiltonians: We give a protocol to test if $H$ is $\epsilon_1$-close to being $s$-sparse (in the Pauli basis) or $\epsilon_2$-far from being $s$-sparse, with $O(s^{6}/(\epsilon_2^2-\epsilon_1^2)^{6})$ queries. For learning up to error $\epsilon$, we show that $O(s^{4}/\epsilon^{8})$ queries suffice. 3. Learning without memory: The learning results stated above have no dependence on $n$, but require $n$-qubit quantum memory. We give subroutines that allow us to learn without memory; increasing the query complexity by a $(\log n)$-factor in the local case and an $n$-factor in the sparse case. 4. Testing without memory: We give a new subroutine called Pauli hashing, which allows one to tolerantly test $s$-sparse Hamiltonians with $O(s^{14}/(\epsilon_2^2-\epsilon_1^2)^{18})$ queries. A key ingredient is showing that $s$-sparse Pauli channels can be tolerantly tested under the diamond norm with $O(s^2/(\epsilon_2-\epsilon_1)^6)$ queries. Along the way, we prove new structural theorems for local and sparse Hamiltonians. We complement our learning results with polynomially weaker lower bounds. Furthermore, our algorithms use short time evolutions and do not assume prior knowledge of the terms in the support of the Pauli spectrum of $H$.

In this paper we develop a fully nonconforming virtual element method (VEM) of arbitrary approximation order for the two dimensional Cahn-Hilliard equation. We carry out the error analysis for the semidiscrete (continuous-in-time) scheme and verify the theoretical convergence result via numerical experiments. We present a fully discrete scheme which uses a convex splitting Runge-Kutta method to discretize in the temporal variable alongside the virtual element spatial discretization.

In the framework of a mixed finite element method, a structure-preserving formulation for incompressible MHD equations with general boundary conditions is proposed. A leapfrog-type temporal scheme fully decouples the fluid part from the Maxwell part by means of staggered discrete time sequences and, in doing so, partially linearizes the system. Conservation and dissipation properties of the formulation before and after the decoupling are analyzed. We demonstrate optimal spatial and second-order temporal error convergence and conservation and dissipation properties of the proposed method using manufactured solutions, and apply it to the benchmark Orszag-Tang and lid-driven cavity test cases.