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Gaussian elimination (GE) is the most used dense linear solver. Error analysis of GE with selected pivoting strategies on well-conditioned systems can focus on studying the behavior of growth factors. Although exponential growth is possible with GE with partial pivoting (GEPP), growth tends to stay much smaller in practice. Support for this behavior was provided recently by Huang and Tikhomirov's average-case analysis of GEPP, which showed GEPP growth factors for Gaussian matrices stay at most polynomial with very high probability. GE with complete pivoting (GECP) has also seen a lot of recent interest, with improvements to both lower and upper bounds on worst-case GECP growth provided by Bisain, Edelman and Urschel in 2023. We are interested in studying how GEPP and GECP behave on the same linear systems as well as studying large growth on particular subclasses of matrices, including orthogonal matrices. Moreover, as a means to better address the question of why large growth is rarely encountered, we further study matrices with a large difference in growth between using GEPP and GECP, and we explore how the smaller growth strategy dominates behavior in a small neighborhood of the initial matrix.

We examine behavior in an experimental collaboration game that incorporates endogenous network formation. The environment is modeled as a generalization of the voluntary contributions mechanism. By varying the information structure in a controlled laboratory experiment, we examine the underlying mechanisms of reciprocity that generate emergent patterns in linking and contribution decisions. Providing players more detailed information about the sharing behavior of others drastically increases efficiency, and positively affects a number of other key outcomes. To understand the driving causes of these changes in behavior we develop and estimate a structural model for actions and small network panels and identify how social preferences affect behavior. We find that the treatment reduces altruism but stimulates reciprocity, helping players coordinate to reach mutually beneficial outcomes. In a set of counterfactual simulations, we show that increasing trust in the community would encourage higher average contributions at the cost of mildly increased free-riding. Increasing overall reciprocity greatly increases collaborative behavior when there is limited information but can backfire in the treatment, suggesting that negative reciprocity and punishment can reduce efficiency. The largest returns would come from an intervention that drives players away from negative and toward positive reciprocity.

Riemannian optimization is concerned with problems, where the independent variable lies on a smooth manifold. There is a number of problems from numerical linear algebra that fall into this category, where the manifold is usually specified by special matrix structures, such as orthogonality or definiteness. Following this line of research, we investigate tools for Riemannian optimization on the symplectic Stiefel manifold. We complement the existing set of numerical optimization algorithms with a Riemannian trust region method tailored to the symplectic Stiefel manifold. To this end, we derive a matrix formula for the Riemannian Hessian under a right-invariant metric. Moreover, we propose a novel retraction for approximating the Riemannian geodesics. Finally, we conduct a comparative study in which we juxtapose the performance of the Riemannian variants of the steepest descent, conjugate gradients, and trust region methods on selected matrix optimization problems that feature symplectic constraints.

Perturbation analysis has emerged as a significant concern across multiple disciplines, with notable advancements being achieved, particularly in the realm of matrices. This study centers on specific aspects pertaining to tensor T-eigenvalues within the context of the tensor-tensor multiplication. Initially, an analytical perturbation analysis is introduced to explore the sensitivity of T-eigenvalues. In the case of third-order tensors featuring square frontal slices, we extend the classical Gershgorin disc theorem and show that all T-eigenvalues are located inside a union of Gershgorin discs. Additionally, we extend the Bauer-Fike theorem to encompass F-diagonalizable tensors and present two modified versions applicable to more general scenarios. The tensor case of the Kahan theorem, which accounts for general perturbations on Hermite tensors, is also investigated. Furthermore, we propose the concept of pseudospectra for third-order tensors based on tensor-tensor multiplication. We develop four definitions that are equivalent under the spectral norm to characterize tensor $\varepsilon$-pseudospectra. Additionally, we present several pseudospectral properties. To provide visualizations, several numerical examples are also provided to illustrate the $\varepsilon$-pseudospectra of specific tensors at different levels.

For organizations to survive and flourish in the long term, innovation and novelty must be continually introduced, which is particularly true in today's rapidly changing world. This raises a variety of ethical and sustainability considerations that seldom receive the attention they deserve. Existing innovation adoption frameworks often focus on technological, organizational, environmental, and social factors impacting adoption. In this chapter, we explore the ethical and sustainability angles, particularly as they relate to emerging technologies, artificial intelligence (AI) being a prominent example. We consider how to facilitate the development and cultivation of innovation cultures in organizations, including budding startups as well as established enterprises, through approaches such as systems thinking.

We propose a set of causal estimands that we call ``the mediated probabilities of causation.'' These estimands quantify the probabilities that an observed negative outcome was induced via a mediating pathway versus a direct pathway in a stylized setting involving a binary exposure or intervention, a single binary mediator, and a binary outcome. We outline a set of conditions sufficient to identify these effects given observed data, and propose a doubly-robust projection based estimation strategy that allows for the use of flexible non-parametric and machine learning methods for estimation. We argue that these effects may be more relevant than the probability of causation, particularly in settings where we observe both some negative outcome and negative mediating event, and we wish to distinguish between settings where the outcome was induced via the exposure inducing the mediator versus the exposure inducing the outcome directly. We motivate our quantities of interest by discussing applications to legal and medical questions of causal attribution.

The importance of exploring a potential integration among surveys has been acknowledged in order to enhance effectiveness and minimize expenses. In this work, we employ the alignment method to combine information from two different surveys for the estimation of complex statistics. The derivation of the alignment weights poses challenges in case of complex statistics due to their non-linear form. To overcome this, we propose to use a linearized variable associated with the complex statistic under consideration. Linearized variables have been widely used to derive variance estimates, thus allowing for the estimation of the variance of the combined complex statistics estimates. Simulations conducted show the effectiveness of the proposed approach, resulting to the reduction of the variance of the combined complex statistics estimates. Also, in some cases, the usage of the alignment weights derived using the linearized variable associated with a complex statistic, could result in a further reduction of the variance of the combined estimates.

The implication problem for conditional independence (CI) asks whether the fact that a probability distribution obeys a given finite set of CI relations implies that a further CI statement also holds in this distribution. This problem has a long and fascinating history, cumulating in positive results about implications now known as the semigraphoid axioms as well as impossibility results about a general finite characterization of CI implications. Motivated by violation of faithfulness assumptions in causal discovery, we study the implication problem in the special setting where the CI relations are obtained from a directed acyclic graphical (DAG) model along with one additional CI statement. Focusing on the Gaussian case, we give a complete characterization of when such an implication is graphical by using algebraic techniques. Moreover, prompted by the relevance of strong faithfulness in statistical guarantees for causal discovery algorithms, we give a graphical solution for an approximate CI implication problem, in which we ask whether small values of one additional partial correlation entail small values for yet a further partial correlation.

Significant recent progress has been made on deriving combination rules that can take as input a set of arbitrarily dependent p-values, and produce as output a single valid p-value. Here, we show that under the assumption of exchangeability of the p-values, many of those rules can be improved (made more powerful). While this observation by itself has practical implications (for example, under repeated tests involving data splitting), it also has implications for combining arbitrarily dependent p-values, since the latter can be made exchangeable by applying a uniformly random permutation. In particular, we derive several simple randomized combination rules for arbitrarily dependent p-values that are more powerful than their deterministic counterparts. For example, we derive randomized and exchangeable improvements of well known p-value combination rules like "twice the median" and "twice the average", as well as geometric and harmonic means. The main technical advance is to show that all these combination rules can be obtained by calibrating the p-values to e-values (using an $\alpha$-dependent calibrator), averaging those e-values, converting to a level $\alpha$ test using Markov's inequality, and finally obtaining p-values by combining this family of tests. The improvements are delivered via recent randomized and exchangeable variants of Markov's inequality.

We propose a method for obtaining parsimonious decompositions of networks into higher order interactions which can take the form of arbitrary motifs.The method is based on a class of analytically solvable generative models, where vertices are connected via explicit copies of motifs, which in combination with non-parametric priors allow us to infer higher order interactions from dyadic graph data without any prior knowledge on the types or frequencies of such interactions. Crucially, we also consider 'degree--corrected' models that correctly reflect the degree distribution of the network and consequently prove to be a better fit for many real world--networks compared to non-degree corrected models. We test the presented approach on simulated data for which we recover the set of underlying higher order interactions to a high degree of accuracy. For empirical networks the method identifies concise sets of atomic subgraphs from within thousands of candidates that cover a large fraction of edges and include higher order interactions of known structural and functional significance. The method not only produces an explicit higher order representation of the network but also a fit of the network to analytically tractable models opening new avenues for the systematic study of higher order network structures.