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We present a rigorous and precise analysis of the maximum degree and the average degree in a dynamic duplication-divergence graph model introduced by Sol\'e, Pastor-Satorras et al. in which the graph grows according to a duplication-divergence mechanism, i.e. by iteratively creating a copy of some node and then randomly alternating the neighborhood of a new node with probability $p$. This model captures the growth of some real-world processes e.g. biological or social networks. In this paper, we prove that for some $0 < p < 1$ the maximum degree and the average degree of a duplication-divergence graph on $t$ vertices are asymptotically concentrated with high probability around $t^p$ and $\max\{t^{2 p - 1}, 1\}$, respectively, i.e. they are within at most a polylogarithmic factor from these values with probability at least $1 - t^{-A}$ for any constant $A > 0$.

Steepest descent methods combining complex contour deformation with numerical quadrature provide an efficient and accurate approach for the evaluation of highly oscillatory integrals. However, unless the phase function governing the oscillation is particularly simple, their application requires a significant amount of a priori analysis and expert user input, to determine the appropriate contour deformation, and to deal with the non-uniformity in the accuracy of standard quadrature techniques associated with the coalescence of stationary points (saddle points) with each other, or with the endpoints of the original integration contour. In this paper we present a novel algorithm for the numerical evaluation of oscillatory integrals with general polynomial phase functions, which automates the contour deformation process and avoids the difficulties typically encountered with coalescing stationary points and endpoints. The inputs to the algorithm are simply the phase and amplitude functions, the endpoints and orientation of the original integration contour, and a small number of numerical parameters. By a series of numerical experiments we demonstrate that the algorithm is accurate and efficient over a large range of frequencies, even for examples with a large number of coalescing stationary points and with endpoints at infinity. As a particular application, we use our algorithm to evaluate cuspoid canonical integrals from scattering theory. A Matlab implementation of the algorithm is made available and is called PathFinder.

Quadratization of polynomial and nonpolynomial systems of ordinary differential equations is advantageous in a variety of disciplines, such as systems theory, fluid mechanics, chemical reaction modeling and mathematical analysis. A quadratization reveals new variables and structures of a model, which may be easier to analyze, simulate, control, and provides a convenient parametrization for learning. This paper presents novel theory, algorithms and software capabilities for quadratization of non-autonomous ODEs. We provide existence results, depending on the regularity of the input function, for cases when a quadratic-bilinear system can be obtained through quadratization. We further develop existence results and an algorithm that generalizes the process of quadratization for systems with arbitrary dimension that retain the nonlinear structure when the dimension grows. For such systems, we provide dimension-agnostic quadratization. An example is semi-discretized PDEs, where the nonlinear terms remain symbolically identical when the discretization size increases. As an important aspect for practical adoption of this research, we extended the capabilities of the QBee software towards both non-autonomous systems of ODEs and ODEs with arbitrary dimension. We present several examples of ODEs that were previously reported in the literature, and where our new algorithms find quadratized ODE systems with lower dimension than the previously reported lifting transformations. We further highlight an important area of quadratization: reduced-order model learning. This area can benefit significantly from working in the optimal lifting variables, where quadratic models provide a direct parametrization of the model that also avoids additional hyperreduction for the nonlinear terms. A solar wind example highlights these advantages.

While many phenomena in physics and engineering are formally high-dimensional, their long-time dynamics often live on a lower-dimensional manifold. The present work introduces an autoencoder framework that combines implicit regularization with internal linear layers and $L_2$ regularization (weight decay) to automatically estimate the underlying dimensionality of a data set, produce an orthogonal manifold coordinate system, and provide the mapping functions between the ambient space and manifold space, allowing for out-of-sample projections. We validate our framework's ability to estimate the manifold dimension for a series of datasets from dynamical systems of varying complexities and compare to other state-of-the-art estimators. We analyze the training dynamics of the network to glean insight into the mechanism of low-rank learning and find that collectively each of the implicit regularizing layers compound the low-rank representation and even self-correct during training. Analysis of gradient descent dynamics for this architecture in the linear case reveals the role of the internal linear layers in leading to faster decay of a "collective weight variable" incorporating all layers, and the role of weight decay in breaking degeneracies and thus driving convergence along directions in which no decay would occur in its absence. We show that this framework can be naturally extended for applications of state-space modeling and forecasting by generating a data-driven dynamic model of a spatiotemporally chaotic partial differential equation using only the manifold coordinates. Finally, we demonstrate that our framework is robust to hyperparameter choices.

We present a finite element approach for diffusion problems with thermal fluctuations based on a fluctuating hydrodynamics model. The governing transport equations are stochastic partial differential equations with a fluctuating forcing term. We propose a discrete formulation of the stochastic forcing term that has the correct covariance matrix up to a standard discretization error. Furthermore, to obtain a numerical solution with spatial correlations that converge to those of the continuum equation, we derive a linear mapping to transform the finite element solution into an equivalent discrete solution that is free from the artificial correlations introduced by the spatial discretization. The method is validated by applying it to two diffusion problems: a second-order diffusion equation and a fourth-order diffusion equation. The theoretical (continuum) solution to the first case presents spatially decorrelated fluctuations, while the second case presents fluctuations correlated over a finite length. In both cases, the numerical solution presents a structure factor that approximates well the continuum one.

In this paper, two novel classes of implicit exponential Runge-Kutta (ERK) methods are studied for solving highly oscillatory systems. First of all, we analyze the symplectic conditions of two kinds of exponential integrators, and present a first-order symplectic method. In order to solve highly oscillatory problems, the highly accurate implicit ERK integrators (up to order four) are formulated by comparing the Taylor expansions of numerical and exact solutions, it is shown that the order conditions of two new kinds of exponential methods are identical to the order conditions of classical Runge-Kutta (RK) methods. Moreover, we investigate the linear stability properties of these exponential methods. Finally, numerical results not only present the long time energy preservation of the first-order symplectic method, but also illustrate the accuracy and efficiency of these formulated methods in comparison with standard ERK methods.

This paper makes the first bridge between the classical differential/boomerang uniformity and the newly introduced $c$-differential uniformity. We show that the boomerang uniformity of an odd APN function is given by the maximum of the entries (except for the first row/column) of the function's $(-1)$-Difference Distribution Table. In fact, the boomerang uniformity of an odd permutation APN function equals its $(-1)$-differential uniformity. We next apply this result to easily compute the boomerang uniformity of several odd APN functions. In the second part we give two classes of differentially low-uniform functions obtained by modifying the inverse function. The first class of permutations (CCZ-inequivalent to the inverse) over a finite field $\mathbb{F}_{p^n}$ ($p$, an odd prime) is obtained from the composition of the inverse function with an order-$3$ cycle permutation, with differential uniformity $3$ if $p=3$ and $n$ is odd; $5$ if $p=13$ and $n$ is even; and $4$ otherwise. The second class is a family of binomials and we show that their differential uniformity equals~$4$. We next complete the open case of $p=3$ in the investigation started by G\" olo\u glu and McGuire (2014), for $p\geq 5$, and continued by K\"olsch (2021), for $p=2$, $n\geq 5$, on the characterization of $L_1(X^{p^n-2})+L_2(X)$ (with linearized $L_1,L_2$) being a permutation polynomial. Finally, we extend to odd characteristic a result of Charpin and Kyureghyan (2010) providing an upper bound for the differential uniformity of the function and its switched version via a trace function.

The objective of this paper is to compute initial conditions for quasi-linear hyperbolic equations. Our proposed approach involves approximating the solution of the hyperbolic equation by truncating its Fourier expansion in the time domain using the polynomial-exponential basis. This truncation enables the elimination of the time variable, resulting in a system of quasi-linear elliptic equations. Thus, we refer to our approach as the "time dimensional reduction method." To solve this system globally without requesting a good initial guess, we employ the Carleman contraction principle. To demonstrate the effectiveness of our method, we provide several numerical examples. The time dimensional reduction method not only provides accurate solutions but also exhibits exceptional computational speed.

We propose a method for computing the Lyapunov exponents of renewal equations (delay equations of Volterra type) and of coupled systems of renewal and delay differential equations. The method consists in the reformulation of the delay equation as an abstract differential equation, the reduction of the latter to a system of ordinary differential equations via pseudospectral collocation, and the application of the standard discrete QR method. The effectiveness of the method is shown experimentally and a MATLAB implementation is provided.

The present article proposes a partitioned Dirichlet-Neumann algorithm, that allows to address unique challenges arising from a novel mixed-dimensional coupling of very slender fibers embedded in fluid flow using a regularized mortar-type finite element discretization. The fibers are modeled via one-dimensional (1D) partial differential equations based on geometrically exact nonlinear beam theory, while the flow is described by the three-dimensional (3D) incompressible Navier-Stokes equations. The arising truly mixed-dimensional 1D-3D coupling scheme constitutes a novel approximate model and numerical strategy, that naturally necessitates specifically tailored solution schemes to ensure an accurate and efficient computational treatment. In particular, we present a strongly coupled partitioned solution algorithm based on a Quasi-Newton method for applications involving fibers with high slenderness ratios that usually present a challenge with regard to the well-known added mass effect. The influence of all employed algorithmic and numerical parameters, namely the applied acceleration technique, the employed constraint regularization parameter as well as shape functions, on efficiency and results of the solution procedure is studied through appropriate examples. Finally, the convergence of the two-way coupled mixed-dimensional problem solution under uniform mesh refinement is demonstrated, a comparison to a 3D reference solution is performed, and the method's capabilities in capturing flow phenomena at large geometric scale separation is illustrated by the example of a submersed vegetation canopy.