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This paper concerns an expansion of first-order Belnap-Dunn logic whose connectives and quantifiers are all familiar from classical logic. The language and logical consequence relation of the logic are defined, a proof system for the defined logic is presented, and the soundness and completeness of the presented proof system is established. The close relationship between the logical consequence relations of the defined logic and the version of classical logic with the same language is illustrated by the minor differences between the presented proof system and a sound and complete proof system for the version of classical logic with the same language. Moreover, fifteen classical laws of logical equivalence are given by which the logical equivalence relation of the defined logic distinguishes itself from the logical equivalence relation of many logics that are closely related at first glance.

We define an asymptotically normal wavelet-based strongly consistent estimator for the Hurst parameter of any Hermite processes. This estimator is obtained by considering a modified wavelet variation in which coefficients are wisely chosen to be, up to negligeable remainders, independent. We use Stein-Malliavin calculus to prove that this wavelet variation satisfies a multidimensional Central Limit Theorem, with an explicit bound for the Wasserstein distance.

Functional Differential Equations (FDEs) play a fundamental role in many areas of mathematical physics, including fluid dynamics (Hopf characteristic functional equation), quantum field theory (Schwinger-Dyson equation), and statistical physics. Despite their significance, computing solutions to FDEs remains a longstanding challenge in mathematical physics. In this paper we address this challenge by introducing new approximation theory and high-performance computational algorithms designed for solving FDEs on tensor manifolds. Our approach involves approximating FDEs using high-dimensional partial differential equations (PDEs), and then solving such high-dimensional PDEs on a low-rank tensor manifold leveraging high-performance parallel tensor algorithms. The effectiveness of the proposed approach is demonstrated through its application to the Burgers-Hopf FDE, which governs the characteristic functional of the stochastic solution to the Burgers equation evolving from a random initial state.

We propose a new concept of lifts of reversible diffusion processes and show that various well-known non-reversible Markov processes arising in applications are lifts in this sense of simple reversible diffusions. Furthermore, we introduce a concept of non-asymptotic relaxation times and show that these can at most be reduced by a square root through lifting, generalising a related result in discrete time. Finally, we demonstrate how the recently developed approach to quantitative hypocoercivity based on space-time Poincar\'e inequalities can be rephrased and simplified in the language of lifts and how it can be applied to find optimal lifts.

Belnap-Dunn logic, also knows as the logic of First-Degree Entailment, is a logic that can serve as the underlying logic of theories that are inconsistent or incomplete. For various reasons, different expansions of Belnap-Dunn logic with non-classical connectives have been studied. This paper investigates the question whether those expansions are interdefinable with an expansion whose connectives include only classical connectives. This is worth knowing because it is difficult to say how close a logic with non-classical connectives is related to classical logic. The notion of interdefinability of logics used is based on a general notion of definability of a connective in a logic that seems to have been forgotten.

A significant hurdle in the noisy intermediate-scale quantum (NISQ) era is identifying functional quantum circuits. These circuits must also adhere to the constraints imposed by current quantum hardware limitations. Variational quantum algorithms (VQAs), a class of quantum-classical optimization algorithms, were developed to address these challenges in the currently available quantum devices. However, the overall performance of VQAs depends on the initialization strategy of the variational circuit, the structure of the circuit (also known as ansatz), and the configuration of the cost function. Focusing on the structure of the circuit, in this thesis, we improve the performance of VQAs by automating the search for an optimal structure for the variational circuits using reinforcement learning (RL). Within the thesis, the optimality of a circuit is determined by evaluating its depth, the overall count of gates and parameters, and its accuracy in solving the given problem. The task of automating the search for optimal quantum circuits is known as quantum architecture search (QAS). The majority of research in QAS is primarily focused on a noiseless scenario. Yet, the impact of noise on the QAS remains inadequately explored. In this thesis, we tackle the issue by introducing a tensor-based quantum circuit encoding, restrictions on environment dynamics to explore the search space of possible circuits efficiently, an episode halting scheme to steer the agent to find shorter circuits, a double deep Q-network (DDQN) with an $\epsilon$-greedy policy for better stability. The numerical experiments on noiseless and noisy quantum hardware show that in dealing with various VQAs, our RL-based QAS outperforms existing QAS. Meanwhile, the methods we propose in the thesis can be readily adapted to address a wide range of other VQAs.

The minimum covariance determinant (MCD) estimator is a popular method for robustly estimating the mean and covariance of multivariate data. We extend the MCD to the setting where the observations are matrices rather than vectors and introduce the matrix minimum covariance determinant (MMCD) estimators for robust parameter estimation. These estimators hold equivariance properties, achieve a high breakdown point, and are consistent under elliptical matrix-variate distributions. We have also developed an efficient algorithm with convergence guarantees to compute the MMCD estimators. Using the MMCD estimators, we can compute robust Mahalanobis distances that can be used for outlier detection. Those distances can be decomposed into outlyingness contributions from each cell, row, or column of a matrix-variate observation using Shapley values, a concept for outlier explanation recently introduced in the multivariate setting. Simulations and examples reveal the excellent properties and usefulness of the robust estimators.

Multiphysics simulations frequently require transferring solution fields between subproblems with non-matching spatial discretizations, typically using interpolation techniques. Standard methods are usually based on measuring the closeness between points by means of the Euclidean distance, which does not account for curvature, cuts, cavities or other non-trivial geometrical or topological features of the domain. This may lead to spurious oscillations in the interpolant in proximity to these features. To overcome this issue, we propose a modification to rescaled localized radial basis function (RL-RBF) interpolation to account for the geometry of the interpolation domain, by yielding conformity and fidelity to geometrical and topological features. The proposed method, referred to as RL-RBF-G, relies on measuring the geodesic distance between data points. RL-RBF-G removes spurious oscillations appearing in the RL-RBF interpolant, resulting in increased accuracy in domains with complex geometries. We demonstrate the effectiveness of RL-RBF-G interpolation through a convergence study in an idealized setting. Furthermore, we discuss the algorithmic aspects and the implementation of RL-RBF-G interpolation in a distributed-memory parallel framework, and present the results of a strong scalability test yielding nearly ideal results. Finally, we show the effectiveness of RL-RBF-G interpolation in multiphysics simulations by considering an application to a whole-heart cardiac electromecanics model.

Chemical and biochemical reactions can exhibit surprisingly different behaviours from multiple steady-state solutions to oscillatory solutions and chaotic behaviours. Such behaviour has been of great interest to researchers for many decades. The Briggs-Rauscher, Belousov-Zhabotinskii and Bray-Liebhafsky reactions, for which periodic variations in concentrations can be visualized by changes in colour, are experimental examples of oscillating behaviour in chemical systems. These type of systems are modelled by a system of partial differential equations coupled by a nonlinearity. However, analysing the pattern, one may suspect that the dynamic is only generated by a finite number of spatial Fourier modes. In fluid dynamics, it is shown that for large times, the solution is determined by a finite number of spatial Fourier modes, called determining modes. In the article, we first introduce the concept of determining modes and show that, indeed, it is sufficient to characterise the dynamic by only a finite number of spatial Fourier modes. In particular, we analyse the exact number of the determining modes of $u$ and $v$, where the couple $(u,v)$ solves the following stochastic system \begin{equation*} \partial_t{u}(t) = r_1\Delta u(t) -\alpha_1u(t)- \gamma_1u(t)v^2(t) + f(1 - u(t)) + g(t),\quad \partial_t{v}(t) = r_2\Delta v(t) -\alpha_2v(t) + \gamma_2 u(t)v^2(t) + h(t),\quad u(0) = u_0,\;v(0) = v_0, \end{equation*} where $r_1,r_2,\gamma_1,\gamma_2>0$, $\alpha_1,\alpha_2 \ge 0$ and $g,h$ are time depending mappings specified later.

We propose an efficient algorithm for matching two correlated Erd\H{o}s--R\'enyi graphs with $n$ vertices whose edges are correlated through a latent vertex correspondence. When the edge density $q= n^{- \alpha+o(1)}$ for a constant $\alpha \in [0,1)$, we show that our algorithm has polynomial running time and succeeds to recover the latent matching as long as the edge correlation is non-vanishing. This is closely related to our previous work on a polynomial-time algorithm that matches two Gaussian Wigner matrices with non-vanishing correlation, and provides the first polynomial-time random graph matching algorithm (regardless of the regime of $q$) when the edge correlation is below the square root of the Otter's constant (which is $\approx 0.338$).