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It is disproved the Tokareva's conjecture that any balanced boolean function of appropriate degree is a derivative of some bent function. This result is based on new upper bounds for the numbers of bent and plateaued functions.

In this paper we establish limit theorems for power variations of stochastic processes controlled by fractional Brownian motions with Hurst parameter $H\leq 1/2$. We show that the power variations of such processes can be decomposed into the mix of several weighted random sums plus some remainder terms, and the convergences of power variations are dominated by different combinations of those weighted sums depending on whether $H<1/4$, $H=1/4$, or $H>1/4$. We show that when $H\geq 1/4$ the centered power variation converges stably at the rate $n^{-1/2}$, and when $H<1/4$ it converges in probability at the rate $n^{-2H}$. We determine the limit of the mixed weighted sum based on a rough path approach developed in \cite{LT20}.

A general class of the almost instantaneous fixed-to-variable-length (AIFV) codes is proposed, which contains every possible binary code we can make when allowing finite bits of decoding delay. The contribution of the paper lies in the following. (i) Introducing $N$-bit-delay AIFV codes, constructed by multiple code trees with higher flexibility than the conventional AIFV codes. (ii) Proving that the proposed codes can represent any uniquely-encodable and uniquely-decodable variable-to-variable length codes. (iii) Showing how to express codes as multiple code trees with minimum decoding delay. (iv) Formulating the constraints of decodability as the comparison of intervals in the real number line. The theoretical results in this paper are expected to be useful for further study on AIFV codes.

A novel overlapping domain decomposition splitting algorithm based on a Crank-Nisolson method is developed for the stochastic nonlinear Schroedinger equation driven by a multiplicative noise with non-periodic boundary conditions. The proposed algorithm can significantly reduce the computational cost while maintaining the similar conservation laws. Numerical experiments are dedicated to illustrating the capability of the algorithm for different spatial dimensions, as well as the various initial conditions. In particular, we compare the performance of the overlapping domain decomposition splitting algorithm with the stochastic multi-symplectic method in [S. Jiang, L. Wang and J. Hong, Commun. Comput. Phys., 2013] and the finite difference splitting scheme in [J. Cui, J. Hong, Z. Liu and W. Zhou, J. Differ. Equ., 2019]. We observe that our proposed algorithm has excellent computational efficiency and is highly competitive. It provides a useful tool for solving stochastic partial differential equations.

Binary responses arise in a multitude of statistical problems, including binary classification, bioassay, current status data problems and sensitivity estimation. There has been an interest in such problems in the Bayesian nonparametrics community since the early 1970s, but inference given binary data is intractable for a wide range of modern simulation-based models, even when employing MCMC methods. Recently, Christensen (2023) introduced a novel simulation technique based on counting permutations, which can estimate both posterior distributions and marginal likelihoods for any model from which a random sample can be generated. However, the accompanying implementation of this technique struggles when the sample size is too large (n > 250). Here we present perms, a new implementation of said technique which is substantially faster and able to handle larger data problems than the original implementation. It is available both as an R package and a Python library. The basic usage of perms is illustrated via two simple examples: a tractable toy problem and a bioassay problem. A more complex example involving changepoint analysis is also considered. We also cover the details of the implementation and illustrate the computational speed gain of perms via a simple simulation study.

This paper introduces a formulation of the variable density incompressible Navier-Stokes equations by modifying the nonlinear terms in a consistent way. For Galerkin discretizations, the formulation leads to full discrete conservation of mass, squared density, momentum, angular momentum and kinetic energy without the divergence-free constraint being strongly enforced. In addition to favorable conservation properties, the formulation is shown to make the density field invariant to global shifts. The effect of viscous regularizations on conservation properties is also investigated. Numerical tests validate the theory developed in this work. The new formulation shows superior performance compared to other formulations from the literature, both in terms of accuracy for smooth problems and in terms of robustness.

In this paper, an optimization problem with uncertain constraint coefficients is considered. Possibility theory is used to model the uncertainty. Namely, a joint possibility distribution in constraint coefficient realizations, called scenarios, is specified. This possibility distribution induces a necessity measure in scenario set, which in turn describes an ambiguity set of probability distributions in scenario set. The distributionally robust approach is then used to convert the imprecise constraints into deterministic equivalents. Namely, the left-hand side of an imprecise constraint is evaluated by using a risk measure with respect to the worst probability distribution that can occur. In this paper, the Conditional Value at Risk is used as the risk measure, which generalizes the strict robust and expected value approaches, commonly used in literature. A general framework for solving such a class of problems is described. Some cases which can be solved in polynomial time are identified.

Cryptographic key exchange protocols traditionally rely on computational conjectures such as the hardness of prime factorisation to provide security against eavesdropping attacks. Remarkably, quantum key distribution protocols like the one proposed by Bennett and Brassard provide information-theoretic security against such attacks, a much stronger form of security unreachable by classical means. However, quantum protocols realised so far are subject to a new class of attacks exploiting implementation defects in the physical devices involved, as demonstrated in numerous ingenious experiments. Following the pioneering work of Ekert proposing the use of entanglement to bound an adversary's information from Bell's theorem, we present here the experimental realisation of a complete quantum key distribution protocol immune to these vulnerabilities. We achieve this by combining theoretical developments on finite-statistics analysis, error correction, and privacy amplification, with an event-ready scheme enabling the rapid generation of high-fidelity entanglement between two trapped-ion qubits connected by an optical fibre link. The secrecy of our key is guaranteed device-independently: it is based on the validity of quantum theory, and certified by measurement statistics observed during the experiment. Our result shows that provably secure cryptography with real-world devices is possible, and paves the way for further quantum information applications based on the device-independence principle.

We prove that every poset with bounded cliquewidth and with sufficiently large dimension contains the standard example of dimension $k$ as a subposet. This applies in particular to posets whose cover graphs have bounded treewidth, as the cliquewidth of a poset is bounded in terms of the treewidth of the cover graph. For the latter posets, we prove a stronger statement: every such poset with sufficiently large dimension contains the Kelly example of dimension $k$ as a subposet. Using this result, we obtain a full characterization of the minor-closed graph classes $\mathcal{C}$ such that posets with cover graphs in $\mathcal{C}$ have bounded dimension: they are exactly the classes excluding the cover graph of some Kelly example. Finally, we consider a variant of poset dimension called Boolean dimension, and we prove that posets with bounded cliquewidth have bounded Boolean dimension. The proofs rely on Colcombet's deterministic version of Simon's factorization theorem, which is a fundamental tool in formal language and automata theory, and which we believe deserves a wider recognition in structural and algorithmic graph theory.

The goal of explainable Artificial Intelligence (XAI) is to generate human-interpretable explanations, but there are no computationally precise theories of how humans interpret AI generated explanations. The lack of theory means that validation of XAI must be done empirically, on a case-by-case basis, which prevents systematic theory-building in XAI. We propose a psychological theory of how humans draw conclusions from saliency maps, the most common form of XAI explanation, which for the first time allows for precise prediction of explainee inference conditioned on explanation. Our theory posits that absent explanation humans expect the AI to make similar decisions to themselves, and that they interpret an explanation by comparison to the explanations they themselves would give. Comparison is formalized via Shepard's universal law of generalization in a similarity space, a classic theory from cognitive science. A pre-registered user study on AI image classifications with saliency map explanations demonstrate that our theory quantitatively matches participants' predictions of the AI.