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The sum of quantum computing errors is the key element both for the estimation and control of errors in quantum computing and for its statistical study. In this article we analyze the sum of two independent quantum computing errors, $X_1$ and $X_2$, and we obtain the formula of the variance of the sum of these errors: $$ V(X_1+X_2)=V(X_1)+V(X_2)-\frac{V(X_1)V(X_2)}{2}. $$ We conjecture that this result holds true for general quantum computing errors and we prove the formula for independent isotropic quantum computing errors.

Accelerated failure time (AFT) models are frequently used to model survival data, providing a direct quantification of the relationship between event times and covariates. These models allow for the acceleration or deceleration of failure times through a multiplicative factor that accounts for the effect of covariates. While existing literature provides numerous methods for fitting AFT models with time-fixed covariates, adapting these approaches to scenarios involving both time-varying covariates and partly interval-censored data remains challenging. Motivated by a randomised clinical trial dataset on advanced melanoma patients, we propose a maximum penalised likelihood approach for fitting a semiparametric AFT model to survival data with partly interval-censored failure times. This method also accommodates both time-fixed and time-varying covariates. We utilise Gaussian basis functions to construct a smooth approximation of the non-parametric baseline hazard and fit the model using a constrained optimisation approach. The effectiveness of our method is demonstrated through extensive simulations. Finally, we illustrate the relevance of our approach by applying it to a dataset from a randomised clinical trial involving patients with advanced melanoma.

Model misspecification analysis strategies, such as anomaly detection, model validation, and model comparison are a key component of scientific model development. Over the last few years, there has been a rapid rise in the use of simulation-based inference (SBI) techniques for Bayesian parameter estimation, applied to increasingly complex forward models. To move towards fully simulation-based analysis pipelines, however, there is an urgent need for a comprehensive simulation-based framework for model misspecification analysis. In this work, we provide a solid and flexible foundation for a wide range of model discrepancy analysis tasks, using distortion-driven model misspecification tests. From a theoretical perspective, we introduce the statistical framework built around performing many hypothesis tests for distortions of the simulation model. We also make explicit analytic connections to classical techniques: anomaly detection, model validation, and goodness-of-fit residual analysis. Furthermore, we introduce an efficient self-calibrating training algorithm that is useful for practitioners. We demonstrate the performance of the framework in multiple scenarios, making the connection to classical results where they are valid. Finally, we show how to conduct such a distortion-driven model misspecification test for real gravitational wave data, specifically on the event GW150914.

In this work we build optimal experimental designs for precise estimation of the functional coefficient of a function-on-function linear regression model where both the response and the factors are continuous functions of time. After obtaining the variance-covariance matrix of the estimator of the functional coefficient which minimizes the integrated sum of square of errors, we extend the classical definition of optimal design to this estimator, and we provide the expression of the A-optimal and of the D-optimal designs. Examples of optimal designs for dynamic experimental factors are then computed through a suitable algorithm, and we discuss different scenarios in terms of the set of basis functions used for their representation. Finally, we present an example with simulated data to illustrate the feasibility of our methodology.

In deep learning, the mean of a chosen error metric, such as squared or absolute error, is commonly used as a loss function. While effective in reducing the average error, this approach often fails to address localized outliers, leading to significant inaccuracies in regions with sharp gradients or discontinuities. This issue is particularly evident in physics-informed neural networks (PINNs), where such localized errors are expected and affect the overall solution. To overcome this limitation, we propose a novel loss function that combines the mean and the standard deviation of the chosen error metric. By minimizing this combined loss function, the method ensures a more uniform error distribution and reduces the impact of localized high-error regions. The proposed loss function was tested on three problems: Burger's equation, 2D linear elastic solid mechanics, and 2D steady Navier-Stokes, demonstrating improved solution quality and lower maximum errors compared to the standard mean-based loss, using the same number of iterations and weight initialization.

The problem of estimating, from a random sample of points, the dimension of a compact subset S of the Euclidean space is considered. The emphasis is put on consistency results in the statistical sense. That is, statements of convergence to the true dimension value when the sample size grows to infinity. Among the many available definitions of dimension, we have focused (on the grounds of its statistical tractability) on three notions: the Minkowski dimension, the correlation dimension and the, perhaps less popular, concept of pointwise dimension. We prove the statistical consistency of some natural estimators of these quantities. Our proofs partially rely on the use of an instrumental estimator formulated in terms of the empirical volume function Vn (r), defined as the Lebesgue measure of the set of points whose distance to the sample is at most r. In particular, we explore the case in which the true volume function V (r) of the target set S is a polynomial on some interval starting at zero. An empirical study is also included. Our study aims to provide some theoretical support, and some practical insights, for the problem of deciding whether or not the set S has a dimension smaller than that of the ambient space. This is a major statistical motivation of the dimension studies, in connection with the so-called Manifold Hypothesis.

A statistical network model with overlapping communities can be generated as a superposition of mutually independent random graphs of varying size. The model is parameterized by the number of nodes, the number of communities, and the joint distribution of the community size and the edge probability. This model admits sparse parameter regimes with power-law limiting degree distributions and non-vanishing clustering coefficients. This article presents large-scale approximations of clique and cycle frequencies for graph samples generated by the model, which are valid for regimes with unbounded numbers of overlapping communities. Our results reveal the growth rates of these subgraph frequencies and show that their theoretical densities can be reliably estimated from data.

We prove, for stably computably enumerable formal systems, direct analogues of the first and second incompleteness theorems of G\"odel. A typical stably computably enumerable set is the set of Diophantine equations with no integer solutions, and in particular such sets are generally not computably enumerable. And so this gives the first extension of the second incompleteness theorem to non classically computable formal systems. Let's motivate this with a somewhat physical application. Let $\mathcal{H} $ be the suitable infinite time limit (stabilization in the sense of the paper) of the mathematical output of humanity, specializing to first order sentences in the language of arithmetic (for simplicity), and understood as a formal system. Suppose that all the relevant physical processes in the formation of $\mathcal{H} $ are Turing computable. Then as defined $\mathcal{H} $ may \emph{not} be computably enumerable, but it is stably computably enumerable. Thus, the classical G\"odel disjunction applied to $\mathcal{H} $ is meaningless, but applying our incompleteness theorems to $\mathcal{H} $ we then get a sharper version of G\"odel's disjunction: assume $\mathcal{H} \vdash PA$ then either $\mathcal{H} $ is not stably computably enumerable or $\mathcal{H} $ is not 1-consistent (in particular is not sound) or $\mathcal{H} $ cannot prove a certain true statement of arithmetic (and cannot disprove it if in addition $\mathcal{H} $ is 2-consistent).

Many combinatorial optimization problems can be formulated as the search for a subgraph that satisfies certain properties and minimizes the total weight. We assume here that the vertices correspond to points in a metric space and can take any position in given uncertainty sets. Then, the cost function to be minimized is the sum of the distances for the worst positions of the vertices in their uncertainty sets. We propose two types of polynomial-time approximation algorithms. The first one relies on solving a deterministic counterpart of the problem where the uncertain distances are replaced with maximum pairwise distances. We study in details the resulting approximation ratio, which depends on the structure of the feasible subgraphs and whether the metric space is Ptolemaic or not. The second algorithm is a fully-polynomial time approximation scheme for the special case of $s-t$ paths.