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Under a multinormal distribution with an arbitrary unknown covariance matrix, the main purpose of this paper is to propose a framework to achieve the goal of reconciliation of Bayesian, frequentist, and Fisher's reporting $p$-values, Neyman-Pearson's optimal theory and Wald's decision theory for the problems of testing mean against restricted alternatives (closed convex cones). To proceed, the tests constructed via the likelihood ratio (LR) and the union-intersection (UI) principles are studied. For the problems of testing against restricted alternatives, first, we show that the LRT and the UIT are not the proper Bayes tests, however, they are shown to be the integrated LRT and the integrated UIT, respectively. For the problem of testing against the positive orthant space alternative, both the null distributions of the LRT and the UIT depend on the unknown nuisance covariance matrix. Hence we have difficulty adopting Fisher's approach to reporting $p$-values. On the other hand, according to the definition of the level of significance, both the LRT and the UIT are shown to be power-dominated by the corresponding LRT and UIT for testing against the half-space alternative, respectively. Hence, both the LRT and the UIT are $\alpha$-inadmissible, these results are against the common statistical sense. Neither Fisher's approach of reporting $p$-values alone nor Neyman-Pearson's optimal theory for power function alone is a satisfactory criterion for evaluating the performance of tests. Wald's decision theory via $d$-admissibility may shed light on resolving these challenging issues of imposing the balance between type 1 error and power.

A recent development in extreme value modeling uses the geometry of the dataset to perform inference on the multivariate tail. A key quantity in this inference is the gauge function, whose values define this geometry. Methodology proposed to date for capturing the gauge function either lacks flexibility due to parametric specifications, or relies on complex neural network specifications in dimensions greater than three. We propose a semiparametric gauge function that is piecewise-linear, making it simple to interpret and provides a good approximation for the true underlying gauge function. This linearity also makes optimization tasks computationally inexpensive. The piecewise-linear gauge function can be used to define both a radial and an angular model, allowing for the joint fitting of extremal pseudo-polar coordinates, a key aspect of this geometric framework. We further expand the toolkit for geometric extremal modeling through the estimation of high radial quantiles at given angular values via kernel density estimation. We apply the new methodology to air pollution data, which exhibits a complex extremal dependence structure.

Many articles have recently been devoted to Mahler equations, partly because of their links with other branches of mathematics such as automata theory. Hahn series (a generalization of the Puiseux series allowing arbitrary exponents of the indeterminate as long as the set that supports them is well-ordered) play a central role in the theory of Mahler equations. In this paper, we address the following fundamental question: is there an algorithm to calculate the Hahn series solutions of a given linear Mahler equation? What makes this question interesting is the fact that the Hahn series appearing in this context can have complicated supports with infinitely many accumulation points. Our (positive) answer to the above question involves among other things the construction of a computable well-ordered receptacle for the supports of the potential Hahn series solutions.

This work is concerned with the computation of the first-order variation for one-dimensional hyperbolic partial differential equations. In the case of shock waves the main challenge is addressed by developing a numerical method to compute the evolution of the generalized tangent vector introduced by Bressan and Marson (1995). Our basic strategy is to combine the conservative numerical schemes and a novel expression of the interface conditions for the tangent vectors along the discontinuity. Based on this, we propose a simple numerical method to compute the tangent vectors for general hyperbolic systems. Numerical results are presented for Burgers' equation and a 2 x 2 hyperbolic system with two genuinely nonlinear fields.

The elapsed time equation is an age-structured model that describes the dynamics of interconnected spiking neurons through the elapsed time since the last discharge, leading to many interesting questions on the evolution of the system from a mathematical and biological point of view. In this work, we first deal with the case when transmission after a spike is instantaneous and the case when there exists a distributed delay that depends on the previous history of the system, which is a more realistic assumption. Then we revisit the well-posedness in order to make a numerical analysis by adapting the classical upwind scheme through a fixed-point approach. We improve the previous results on well-posedness by relaxing some hypotheses on the non-linearity for instantaneous transmission, including the strongly excitatory case, while for the numerical analysis we prove that the approximation given by the explicit upwind scheme converges to the solution of the non-linear problem through BV-estimates. We also show some numerical simulations to compare the behavior of the system in the case of instantaneous transmission with the case of distributed delay under different parameters, leading to solutions with different asymptotic profiles.

We show that one-way functions exist if and only there exists an efficient distribution relative to which almost-optimal compression is hard on average. The result is obtained by combining a theorem of Ilango, Ren, and Santhanam and one by Bauwens and Zimand.

We carry out a stability and convergence analysis for the fully discrete scheme obtained by combining a finite or virtual element spatial discretization with the upwind-discontinuous Galerkin time-stepping applied to the time-dependent advection-diffusion equation. A space-time streamline-upwind Petrov-Galerkin term is used to stabilize the method. More precisely, we show that the method is inf-sup stable with constant independent of the diffusion coefficient, which ensures the robustness of the method in the convection- and diffusion-dominated regimes. Moreover, we prove optimal convergence rates in both regimes for the error in the energy norm. An important feature of the presented analysis is the control in the full $L^2(0,T;L^2(\Omega))$ norm without the need of introducing an artificial reaction term in the model. We finally present some numerical experiments in $(3 + 1)$-dimensions that validate our theoretical results.

We present explicit representations in terms of hypergeometric functions for the scaling functions in the $C^0$ orthogonal multiresolution analyses associated with piecewise continuous polynomials. Closed formulas for the Mellin transform of these functions as well as their Fourier transforms are derived. Some new multiresolution analyses whose scaling functions have coefficients that are rational numbers are introduced and discussed.

Mathematical modeling is a powerful tool for describing, predicting, and understanding complex phenomena exhibited by real-world systems. However, identifying the equations that govern a system's dynamics from experimental data remains a significant challenge without a definitive solution. In this study, evolutionary computing techniques are presented to estimate the governing equations of a dynamical system using time-series data. The main approach is to propose polynomial equations with unknown coefficients, and subsequently perform a parametric estimation using genetic algorithms. Some of the main contributions of the present study are an adequate modification of the genetic algorithm to remove terms with minimal contributions, and a mechanism to escape local optima during the search. To evaluate the proposed method, we applied it to three dynamical systems: a linear model, a nonlinear model, and the Lorenz system. Our results demonstrate a reconstruction with an Integral Square Error below 0.22 and a coefficient of determination R-squared of 0.99 for all systems, indicating successful reconstruction of the governing dynamic equations.

In Bayesian inverse problems, it is common to consider several hyperparameters that define the prior and the noise model that must be estimated from the data. In particular, we are interested in linear inverse problems with additive Gaussian noise and Gaussian priors defined using Mat\'{e}rn covariance models. In this case, we estimate the hyperparameters using the maximum a posteriori (MAP) estimate of the marginalized posterior distribution. However, this is a computationally intensive task since it involves computing log determinants. To address this challenge, we consider a stochastic average approximation (SAA) of the objective function and use the preconditioned Lanczos method to compute efficient approximations of the function and gradient evaluations. We propose a new preconditioner that can be updated cheaply for new values of the hyperparameters and an approach to compute approximations of the gradient evaluations, by reutilizing information from the function evaluations. We demonstrate the performance of our approach on static and dynamic seismic tomography problems.