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Faces play a central role in the combinatorial and computational aspects of polyhedra. In this paper, we present the first formalization of faces of polyhedra in the proof assistant Coq. This builds on the formalization of a library providing the basic constructions and operations over polyhedra, including projections, convex hulls and images under linear maps. Moreover, we design a special mechanism which automatically introduces an appropriate representation of a polyhedron or a face, depending on the context of the proof. We demonstrate the usability of this approach by establishing some of the most important combinatorial properties of faces, namely that they constitute a family of graded atomistic and coatomistic lattices closed under interval sublattices. We also prove a theorem due to Balinski on the $d$-connectedness of the adjacency graph of polytopes of dimension $d$.

The number of down-steps between pairs of up-steps in $k_t$-Dyck paths, a generalization of Dyck paths consisting of steps $\{(1, k), (1, -1)\}$ such that the path stays (weakly) above the line $y=-t$, is studied. Results are proved bijectively and by means of generating functions, and lead to several interesting identities as well as links to other combinatorial structures. In particular, there is a connection between $k_t$-Dyck paths and perforation patterns for punctured convolutional codes (binary matrices) used in coding theory. Surprisingly, upon restriction to usual Dyck paths this yields a new combinatorial interpretation of Catalan numbers.

The Koopman operator is beneficial for analyzing nonlinear and stochastic dynamics; it is linear but infinite-dimensional, and it governs the evolution of observables. The extended dynamic mode decomposition (EDMD) is one of the famous methods in the Koopman operator approach. The EDMD employs a data set of snapshot pairs and a specific dictionary to evaluate an approximation for the Koopman operator, i.e., the Koopman matrix. In this study, we focus on stochastic differential equations, and a method to obtain the Koopman matrix is proposed. The proposed method does not need any data set, which employs the original system equations to evaluate some of the targeted elements of the Koopman matrix. The proposed method comprises combinatorics, an approximation of the resolvent, and extrapolations. Comparisons with the EDMD are performed for a noisy van der Pol system. The proposed method yields reasonable results even in cases wherein the EDMD exhibits a slow convergence behavior.

We study dynamic algorithms for the problem of maximizing a monotone submodular function over a stream of $n$ insertions and deletions. We show that any algorithm that maintains a $(0.5+\epsilon)$-approximate solution under a cardinality constraint, for any constant $\epsilon>0$, must have an amortized query complexity that is $\mathit{polynomial}$ in $n$. Moreover, a linear amortized query complexity is needed in order to maintain a $0.584$-approximate solution. This is in sharp contrast with recent dynamic algorithms of [LMNF+20, Mon20] that achieve $(0.5-\epsilon)$-approximation with a $\mathsf{poly}\log(n)$ amortized query complexity. On the positive side, when the stream is insertion-only, we present efficient algorithms for the problem under a cardinality constraint and under a matroid constraint with approximation guarantee $1-1/e-\epsilon$ and amortized query complexities $\smash{O(\log (k/\epsilon)/\epsilon^2)}$ and $\smash{k^{\tilde{O}(1/\epsilon^2)}\log n}$, respectively, where $k$ denotes the cardinality parameter or the rank of the matroid.

This paper addresses the numerical solution of nonlinear eigenvector problems such as the Gross-Pitaevskii and Kohn-Sham equation arising in computational physics and chemistry. These problems characterize critical points of energy minimization problems on the infinite-dimensional Stiefel manifold. To efficiently compute minimizers, we propose a novel Riemannian gradient descent method induced by an energy-adaptive metric. Quantified convergence of the methods is established under suitable assumptions on the underlying problem. A non-monotone line search and the inexact evaluation of Riemannian gradients substantially improve the overall efficiency of the method. Numerical experiments illustrate the performance of the method and demonstrates its competitiveness with well-established schemes.

This study clarifies the proper criteria to assess the modeling capacity of a general tensor model. The work analyze the problem based on the study of tensor ranks, which is not a well-defined quantity for higher order tensors. To process, the author introduces the separability issue to discuss the Cannikin's law of tensor modeling. Interestingly, a connection between entanglement studied in information theory and tensor analysis is established, shedding new light on the theoretical understanding for modeling capacity problems.

This extensive revision of my paper "Description of an $O(\text{poly}(n))$ Algorithm for NP-Complete Combinatorial Problems" will dramatically simplify the content of the original paper by solving subset-sum instead of $3$-SAT. I will first define the "product-derivative" method which will be used to generate a system of equations for solving unknown polynomial coefficients. Then I will describe the "Dragonfly" algorithm usable to solve subset-sum in $O(n^{16}\log(n))$ which is itself composed of a set of symbolic algebra steps on monic polynomials to convert a subset, $S_T$, of a set of positive integers, $S$, with a given target sum, $T$ into a polynomial with roots corresponding to the elements of $S_T$.

The minimum energy path (MEP) describes the mechanism of reaction, and the energy barrier along the path can be used to calculate the reaction rate in thermal systems. The nudged elastic band (NEB) method is one of the most commonly used schemes to compute MEPs numerically. It approximates an MEP by a discrete set of configuration images, where the discretization size determines both computational cost and accuracy of the simulations. In this paper, we consider a discrete MEP to be a stationary state of the NEB method and prove an optimal convergence rate of the discrete MEP with respect to the number of images. Numerical simulations for the transitions of some several proto-typical model systems are performed to support the theory.

"There and Back Again" (TABA) is a programming pattern where the recursive calls traverse one data structure and the subsequent returns traverse another. This article presents new TABA examples, refines existing ones, and formalizes both their control flow and their data flow using the Coq Proof Assistant. Each formalization mechanizes a pen-and-paper proof, thus making it easier to "get" TABA. In addition, this article identifies and illustrates a tail-recursive variant of TABA, There and Forth Again (TAFA) that does not come back but goes forth instead with more tail calls.

Holonomic functions play an essential role in Computer Algebra since they allow the application of many symbolic algorithms. Among all algorithmic attempts to find formulas for power series, the holonomic property remains the most important requirement to be satisfied by the function under consideration. The targeted functions mainly summarize that of meromorphic functions. However, expressions like $\tan(z)$, $z/(\exp(z)-1)$, $\sec(z)$, etc., particularly, reciprocals, quotients and compositions of holonomic functions, are generally not holonomic. Therefore their power series are inaccessible by the holonomic framework. From the mathematical dictionaries, one can observe that most of the known closed-form formulas of non-holonomic power series involve another sequence whose evaluation depends on some finite summations. In the case of $\tan(z)$ and $\sec(z)$ the corresponding sequences are the Bernoulli and Euler numbers, respectively. Thus providing a symbolic approach that yields complete representations when linear summations for power series coefficients of non-holonomic functions appear, might be seen as a step forward towards the representation of non-holonomic power series. By adapting the method of ansatz with undetermined coefficients, we build an algorithm that computes least-order quadratic differential equations with polynomial coefficients for a large class of non-holonomic functions. A differential equation resulting from this procedure is converted into a recurrence equation by applying the Cauchy product formula and rewriting powers into polynomials and derivatives into shifts. Finally, using enough initial values we are able to give normal form representations to characterize several non-holonomic power series and prove non-trivial identities. We discuss this algorithm and its implementation for Maple 2022.