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In this manuscript, we study the stability of the origin for the multivariate geometric Brownian motion. More precisely, under suitable sufficient conditions, we construct a Lyapunov function such that the origin of the multivariate geometric Brownian motion is globally asymptotically stable in probability. Moreover, we show that such conditions can be rewritten as a Bilinear Matrix Inequality (BMI) feasibility problem. We stress that no commutativity relations between the drift matrix and the noise dispersion matrices are assumed and therefore the so-called Magnus representation of the solution of the multivariate geometric Brownian motion is complicated. In addition, we exemplify our method in numerous specific models from the literature such as random linear oscillators, satellite dynamics, inertia systems, diagonal noise systems, cancer self-remission and smoking.

We address the problem of the best uniform approximation of a continuous function on a convex domain. The approximation is by linear combinations of a finite system of functions (not necessarily Chebyshev) under arbitrary linear constraints. By modifying the concept of alternance and of the Remez iterative procedure we present a method, which demonstrates its efficiency in numerical problems. The linear rate of convergence is proved under some favourable assumptions. A special attention is paid to systems of complex exponents, Gaussian functions, lacunar algebraic and trigonometric polynomials. Applications to signal processing, linear ODE, switching dynamical systems, and to Markov-Bernstein type inequalities are considered.

Surface parameterization plays a fundamental role in many science and engineering problems. In particular, as genus-0 closed surfaces are topologically equivalent to a sphere, many spherical parameterization methods have been developed over the past few decades. However, in practice, mapping a genus-0 closed surface onto a sphere may result in a large distortion due to their geometric difference. In this work, we propose a new framework for computing ellipsoidal conformal and quasi-conformal parameterizations of genus-0 closed surfaces, in which the target parameter domain is an ellipsoid instead of a sphere. By combining simple conformal transformations with different types of quasi-conformal mappings, we can easily achieve a large variety of ellipsoidal parameterizations with their bijectivity guaranteed by quasi-conformal theory. Numerical experiments are presented to demonstrate the effectiveness of the proposed framework.

In this paper, we present a discrete formulation of nonlinear shear- and torsion-free rods introduced by Gebhardt and Romero in [20] that uses isogeometric discretization and robust time integration. Omitting the director as an independent variable field, we reduce the number of degrees of freedom and obtain discrete solutions in multiple copies of the Euclidean space (R^3), which is larger than the corresponding multiple copies of the manifold (R^3 x S^2) obtained with standard Hermite finite elements. For implicit time integration, we choose the same integration scheme as Gebhardt and Romero in [20] that is a hybrid form of the midpoint and the trapezoidal rules. In addition, we apply a recently introduced approach for outlier removal by Hiemstra et al. [26] that reduces high-frequency content in the response without affecting the accuracy, ensuring robustness of our nonlinear discrete formulation. We illustrate the efficiency of our nonlinear discrete formulation for static and transient rods under different loading conditions, demonstrating good accuracy in space, time and the frequency domain. Our numerical example coincides with a relevant application case, the simulation of mooring lines.

We introduce a 2-dimensional stochastic dominance (2DSD) index to characterize both strict and almost stochastic dominance. Based on this index, we derive an estimator for the minimum violation ratio (MVR), also known as the critical parameter, of the almost stochastic ordering condition between two variables. We determine the asymptotic properties of the empirical 2DSD index and MVR for the most frequently used stochastic orders. We also provide conditions under which the bootstrap estimators of these quantities are strongly consistent. As an application, we develop consistent bootstrap testing procedures for almost stochastic dominance. The performance of the tests is checked via simulations and the analysis of real data.

We consider the problem of estimating log-determinants of large, sparse, positive definite matrices. A key focus of our algorithm is to reduce computational cost, and it is based on sparse approximate inverses. The algorithm can be implemented to be adaptive, and it uses graph spline approximation to improve accuracy. We illustrate our approach on classes of large sparse matrices.

The amount of information in satisfiability problem (SAT) is considered. SAT can be polynomial-time solvable when the solving algorithm holds an exponential amount of information. It is also established that SAT Kolmogorov complexity is constant. It is argued that the amount of information in SAT grows at least exponentially with the size of the input instance. The amount of information in SAT is compared with the amount of information in the fixed code algorithms and generated over runtime.

Inverse problems, particularly those governed by Partial Differential Equations (PDEs), are prevalent in various scientific and engineering applications, and uncertainty quantification (UQ) of solutions to these problems is essential for informed decision-making. This second part of a two-paper series builds upon the foundation set by the first part, which introduced CUQIpy, a Python software package for computational UQ in inverse problems using a Bayesian framework. In this paper, we extend CUQIpy's capabilities to solve PDE-based Bayesian inverse problems through a general framework that allows the integration of PDEs in CUQIpy, whether expressed natively or using third-party libraries such as FEniCS. CUQIpy offers concise syntax that closely matches mathematical expressions, streamlining the modeling process and enhancing the user experience. The versatility and applicability of CUQIpy to PDE-based Bayesian inverse problems are demonstrated on examples covering parabolic, elliptic and hyperbolic PDEs. This includes problems involving the heat and Poisson equations and application case studies in electrical impedance tomography and photo-acoustic tomography, showcasing the software's efficiency, consistency, and intuitive interface. This comprehensive approach to UQ in PDE-based inverse problems provides accessibility for non-experts and advanced features for experts.

In this paper we present a new "external verifier" for the Lean theorem prover, written in Lean itself. This is the first complete verifier for Lean 4 other than the reference implementation in C++ used by Lean itself, and our new verifier is competitive with the original, running between 20% and 50% slower and usable to verify all of Lean's mathlib library, forming an additional step in Lean's aim to self-host the full elaborator and compiler. Moreover, because the verifier is written in a language which admits formal verification, it is possible to state and prove properties about the kernel itself, and we report on some initial steps taken in this direction to formalize the Lean type theory abstractly and show that the kernel correctly implements this theory, to eliminate the possibility of implementation bugs in the kernel and increase the trustworthiness of proofs conducted in it. This work is still ongoing but we plan to use this project to help justify any future changes to the kernel and type theory and ensure unsoundness does not sneak in through either the abstract theory or implementation bugs.

Printing custom DNA sequences is essential to scientific and biomedical research, but the technology can be used to manufacture plagues as well as cures. Just as ink printers recognize and reject attempts to counterfeit money, DNA synthesizers and assemblers should deny unauthorized requests to make viral DNA that could be used to ignite a pandemic. There are three complications. First, we don't need to quickly update printers to deal with newly discovered currencies, whereas we regularly learn of new viruses and other biological threats. Second, anti-counterfeiting specifications on a local printer can't be extracted and misused by malicious actors, unlike information on biological threats. Finally, any screening must keep the inspected DNA sequences private, as they may constitute valuable trade secrets. Here we describe SecureDNA, a free, privacy-preserving, and fully automated system capable of verifiably screening all DNA synthesis orders of 30+ base pairs against an up-to-date database of hazards, and its operational performance and specificity when applied to 67 million base pairs of DNA synthesized by providers in the United States, Europe, and China.