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The membership and threshold problems for recurrence sequences are fundamental open decision problems in automated verification. The former problem asks whether a chosen target is an element of a sequence, whilst the latter asks whether every term in a sequence is bounded from below by a given value. A rational-valued sequence $\langle u_n \rangle_n$ is hypergeometric if it satisfies a first-order linear recurrence of the form $p(n)u_{n+1} = q(n)u_{n}$ with polynomial coefficients $p,q\in\mathbb{Z}[x]$. In this note we establish decidability results for the aforementioned problems for restricted classes of hypergeometric sequences. For example, we establish decidability for the aforementioned problems under the assumption that the polynomial coefficients $p,q\in\mathbb{Z}[x]$ are monic and split over an imaginary rational extension of $\mathbb{Q}$. We also establish conditional decidability results; that is, conditional on Schanuel's conjecture, when the irreducible factors of the monic polynomial coefficients $p,q\in\mathbb{Z}[x]$ are either linear or quadratic.

We consider a finite-dimensional vector space $W\subset K^E$ over an arbitrary field $K$ and an arbitrary set $E$. We show that the set $C(W)\subset 2^E$ consisting of the minimal supports of $W$ are the circuits of a matroid on $E$. In particular, we show that this matroid is cofinitary (hence, tame). When the cardinality of $K$ is large enough (with respect to the cardinality of $E$), then the set $trop(W)\subset 2^E$ consisting of all the supports of $W$ is a matroid itself. Afterwards we apply these results to tropical differential algebraic geometry and study the set of supports $trop(Sol(\Sigma))\subset (2^{\mathbb{N}^{m}})^n$ of spaces of formal power series solutions $\text{Sol}(\Sigma)$ of systems of linear differential equations $\Sigma$ in differential variables $x_1,\ldots,x_n$ having coefficients in the ring ${K}[\![t_1,\ldots,t_m]\!]$. If $\Sigma$ is of differential type zero, then the set $C(Sol(\Sigma))\subset (2^{\mathbb{N}^{m}})^n$ of minimal supports defines a matroid on $E=\mathbb{N}^{mn}$, and if the cardinality of $K$ is large enough, then the set of supports $trop(Sol(\Sigma))$ itself is a matroid on $E$ as well. By applying the fundamental theorem of tropical differential algebraic geometry (fttdag), we give a necessary condition under which the set of solutions $Sol(U)$ of a system $U$ of tropical linear differential equations to be a matroid. We also give a counterexample to the fttdag for systems $\Sigma$ of linear differential equations over countable fields. In this case, the set $trop(Sol(\Sigma))$ may not form a matroid.

Motivated by recent works on streaming algorithms for constraint satisfaction problems (CSPs), we define and analyze oblivious algorithms for the Max-$k$AND problem. This generalizes the definition by Feige and Jozeph (Algorithmica '15) of oblivious algorithms for Max-DICUT, a special case of Max-$2$AND. Oblivious algorithms round each variable with probability depending only on a quantity called the variable's bias. For each oblivious algorithm, we design a so-called "factor-revealing linear program" (LP) which captures its worst-case instance, generalizing one of Feige and Jozeph for Max-DICUT. Then, departing from their work, we perform a fully explicit analysis of these (infinitely many!) LPs. In particular, we show that for all $k$, oblivious algorithms for Max-$k$AND provably outperform a special subclass of algorithms we call "superoblivious" algorithms. Our result has implications for streaming algorithms: Generalizing the result for Max-DICUT of Saxena, Singer, Sudan, and Velusamy (SODA'23), we prove that certain separation results hold between streaming models for infinitely many CSPs: for every $k$, $O(\log n)$-space sketching algorithms for Max-$k$AND known to be optimal in $o(\sqrt n)$-space can be beaten in (a) $O(\log n)$-space under a random-ordering assumption, and (b) $O(n^{1-1/k} D^{1/k})$ space under a maximum-degree-$D$ assumption. Even in the previously-known case of Max-DICUT, our analytic proof gives a fuller, computer-free picture of these separation results.

Profile likelihoods are rarely used in geostatistical models due to the computational burden imposed by repeated decompositions of large variance matrices. Accounting for uncertainty in covariance parameters can be highly consequential in geostatistical models as some covariance parameters are poorly identified, the problem is severe enough that the differentiability parameter of the Matern correlation function is typically treated as fixed. The problem is compounded with anisotropic spatial models as there are two additional parameters to consider. In this paper, we make the following contributions: 1, A methodology is created for profile likelihoods for Gaussian spatial models with Mat\'ern family of correlation functions, including anisotropic models. This methodology adopts a novel reparametrization for generation of representative points, and uses GPUs for parallel profile likelihoods computation in software implementation. 2, We show the profile likelihood of the Mat\'ern shape parameter is often quite flat but still identifiable, it can usually rule out very small values. 3, Simulation studies and applications on real data examples show that profile-based confidence intervals of covariance parameters and regression parameters have superior coverage to the traditional standard Wald type confidence intervals.

Time-dependent basis reduced order models (TDB ROMs) have successfully been used for approximating the solution to nonlinear stochastic partial differential equations (PDEs). For many practical problems of interest, discretizing these PDEs results in massive matrix differential equations (MDEs) that are too expensive to solve using conventional methods. While TDB ROMs have the potential to significantly reduce this computational burden, they still suffer from the following challenges: (i) inefficient for general nonlinearities, (ii) intrusive implementation, (iii) ill-conditioned in the presence of small singular values, and (iv) error accumulation due to fixed rank. To this end, we present a scalable method for solving TDB ROMs that is computationally efficient, minimally intrusive, robust in the presence of small singular values, rank-adaptive, and highly parallelizable. These favorable properties are achieved via low-rank approximation of the time discrete MDE. Using the discrete empirical interpolation method (DEIM), a low-rank CUR decomposition is computed at each iteration of the time stepping scheme, enabling a near-optimal approximation at a fraction of the cost. We also propose a rank-adaptive procedure to control the error on-the-fly. Numerical results demonstrate the accuracy, efficiency, and robustness of the new method for a diverse set of problems.

A least-squares neural network (LSNN) method was introduced for solving scalar linear and nonlinear hyperbolic conservation laws (HCLs) in [7, 6]. This method is based on an equivalent least-squares (LS) formulation and uses ReLU neural network as approximating functions, making it ideal for approximating discontinuous functions with unknown interface location. In the design of the LSNN method for HCLs, the numerical approximation of differential operators is a critical factor, and standard numerical or automatic differentiation along coordinate directions can often lead to a failed NN-based method. To overcome this challenge, this paper rewrites HCLs in their divergence form of space and time and introduces a new discrete divergence operator. As a result, the proposed LSNN method is free of penalization of artificial viscosity. Theoretically, the accuracy of the discrete divergence operator is estimated even for discontinuous solutions. Numerically, the LSNN method with the new discrete divergence operator was tested for several benchmark problems with both convex and non-convex fluxes, and was able to compute the correct physical solution for problems with rarefaction, shock or compound waves. The method is capable of capturing the shock of the underlying problem without oscillation or smearing, even without any penalization of the entropy condition, total variation, and/or artificial viscosity.

Many real-world systems modeled using differential equations involve unknown or uncertain parameters. Standard approaches to address parameter estimation inverse problems in this setting typically focus on estimating constants; yet some unobservable system parameters may vary with time without known evolution models. In this work, we propose a novel approximation method inspired by the Fourier series to estimate time-varying parameters in deterministic dynamical systems modeled with ordinary differential equations. Using ensemble Kalman filtering in conjunction with Fourier series-based approximation models, we detail two possible implementation schemes for sequentially updating the time-varying parameter estimates given noisy observations of the system states. We demonstrate the capabilities of the proposed approach in estimating periodic parameters, both when the period is known and unknown, as well as non-periodic time-varying parameters of different forms with several computed examples using a forced harmonic oscillator. Results emphasize the importance of the frequencies and number of approximation model terms on the time-varying parameter estimates and corresponding dynamical system predictions.

For terminal value problems of fractional differential equations of order $\alpha \in (0,1)$ that use Caputo derivatives, shooting methods are a well developed and investigated approach. Based on recently established analytic properties of such problems, we develop a new technique to select the required initial values that solves such shooting problems quickly and accurately. Numerical experiments indicate that this new proportional secting technique converges very quickly and accurately to the solution. Run time measurements indicate a speedup factor of between 4 and 10 when compared to the standard bisection method.

It is known that standard stochastic Galerkin methods encounter challenges when solving partial differential equations with high dimensional random inputs, which are typically caused by the large number of stochastic basis functions required. It becomes crucial to properly choose effective basis functions, such that the dimension of the stochastic approximation space can be reduced. In this work, we focus on the stochastic Galerkin approximation associated with generalized polynomial chaos (gPC), and explore the gPC expansion based on the analysis of variance (ANOVA) decomposition. A concise form of the gPC expansion is presented for each component function of the ANOVA expansion, and an adaptive ANOVA procedure is proposed to construct the overall stochastic Galerkin system. Numerical results demonstrate the efficiency of our proposed adaptive ANOVA stochastic Galerkin method.

We investigate random matrices whose entries are obtained by applying a nonlinear kernel function to pairwise inner products between $n$ independent data vectors, drawn uniformly from the unit sphere in $\mathbb{R}^d$. This study is motivated by applications in machine learning and statistics, where these kernel random matrices and their spectral properties play significant roles. We establish the weak limit of the empirical spectral distribution of these matrices in a polynomial scaling regime, where $d, n \to \infty$ such that $n / d^\ell \to \kappa$, for some fixed $\ell \in \mathbb{N}$ and $\kappa \in (0, \infty)$. Our findings generalize an earlier result by Cheng and Singer, who examined the same model in the linear scaling regime (with $\ell = 1$). Our work reveals an equivalence principle: the spectrum of the random kernel matrix is asymptotically equivalent to that of a simpler matrix model, constructed as a linear combination of a (shifted) Wishart matrix and an independent matrix sampled from the Gaussian orthogonal ensemble. The aspect ratio of the Wishart matrix and the coefficients of the linear combination are determined by $\ell$ and the expansion of the kernel function in the orthogonal Hermite polynomial basis. Consequently, the limiting spectrum of the random kernel matrix can be characterized as the free additive convolution between a Marchenko-Pastur law and a semicircle law. We also extend our results to cases with data vectors sampled from isotropic Gaussian distributions instead of spherical distributions.