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We consider the Cauchy problem for the Helmholtz equation with a domain in R^d, d>2 with N cylindrical outlets to infinity with bounded inclusions in R^{d-1}. Cauchy data are prescribed on the boundary of the bounded domains and the aim is to find solution on the unbounded part of the boundary. In 1989, Kozlov and Maz'ya proposed an alternating iterative method for solving Cauchy problems associated with elliptic,self-adjoint and positive-definite operators in bounded domains. Different variants of this method for solving Cauchy problems associated with Helmholtz-type operators exists. We consider the variant proposed by Mpinganzima et al. for bounded domains and derive the necessary conditions for the convergence of the procedure in unbounded domains. For the numerical implementation, a finite difference method is used to solve the problem in a simple rectangular domain in R^2 that represent a truncated infinite strip. The numerical results shows that by appropriate truncation of the domain and with appropriate choice of the Robin parameters, the Robin-Dirichlet alternating iterative procedure is convergent.

Let $\sigma$ be a first-order signature and let $\mathbf{W}_n$ be the set of all $\sigma$-structures with domain $[n] = \{1, \ldots, n\}$. We can think of each structure in $\mathbf{W}_n$ as representing a "possible (state of the) world". By an inference framework we mean a class $\mathbf{F}$ of pairs $(\mathbb{P}, L)$, where $\mathbb{P} = (\mathbb{P}_n : n = 1, 2, 3, \ldots)$ and each $\mathbb{P}_n$ is a probability distribution on $\mathbb{W}_n$, and $L$ is a logic with truth values in the unit interval $[0, 1]$. From the point of view of probabilistic and logical expressivity one may consider an inference framework as optimal if it allows any pair $(\mathbb{P}, L)$ where $\mathbb{P} = (\mathbb{P}_n : n = 1, 2, 3, \ldots)$ is a sequence of probability distributions on $\mathbb{W}_n$ and $L$ is a logic. But from the point of view of using a pair $(\mathbb{P}, L)$ from such an inference framework for making inferences on $\mathbb{W}_n$ when $n$ is large we face the problem of computational complexity. This motivates looking for an "optimal" trade-off (in a given context) between expressivity and computational efficiency. We define a notion that an inference framework is "asymptotically at least as expressive" as another inference framework. This relation is a preorder and we describe a (strict) partial order on the equivalence classes of some inference frameworks that in our opinion are natural in the context of machine learning and artificial intelligence. The results have bearing on issues concerning efficient learning and probabilistic inference, but are also new instances of results in finite model theory about "almost sure elimination" of extra syntactic features (e.g quantifiers) beyond the connectives. Often such a result has a logical convergence law as a corollary.

In a sports competition, a team might lose a powerful incentive to exert full effort if its final rank does not depend on the outcome of the matches still to be played. Therefore, the organiser should reduce the probability of such a situation to the extent possible. Our paper provides a classification scheme to identify these weakly (where one team is indifferent) or strongly (where both teams are indifferent) stakeless games. A statistical model is estimated to simulate the UEFA Champions League groups and compare the candidate schedules used in the 2021/22 season according to the competitiveness of the matches played in the last round(s). The option followed in four of the eight groups is found to be optimal under a wide set of parameters. Minimising the number of strongly stakeless matches is verified to be a likely goal in the computer draw of the fixture that remains hidden from the public.

We study the problem of testing whether a function $f: \mathbb{R}^n \to \mathbb{R}$ is a polynomial of degree at most $d$ in the \emph{distribution-free} testing model. Here, the distance between functions is measured with respect to an unknown distribution $\mathcal{D}$ over $\mathbb{R}^n$ from which we can draw samples. In contrast to previous work, we do not assume that $\mathcal{D}$ has finite support. We design a tester that given query access to $f$, and sample access to $\mathcal{D}$, makes $(d/\varepsilon)^{O(1)}$ many queries to $f$, accepts with probability $1$ if $f$ is a polynomial of degree $d$, and rejects with probability at least $2/3$ if every degree-$d$ polynomial $P$ disagrees with $f$ on a set of mass at least $\varepsilon$ with respect to $\mathcal{D}$. Our result also holds under mild assumptions when we receive only a polynomial number of bits of precision for each query to $f$, or when $f$ can only be queried on rational points representable using a logarithmic number of bits. Along the way, we prove a new stability theorem for multivariate polynomials that may be of independent interest.

Let ${\mathcal M}\subset {\mathbb R}^n$ be a $C^2$-smooth compact submanifold of dimension $d$. Assume that the volume of ${\mathcal M}$ is at most $V$ and the reach (i.e. the normal injectivity radius) of ${\mathcal M}$ is greater than $\tau$. Moreover, let $\mu$ be a probability measure on ${\mathcal M}$ whose density on ${\mathcal M}$ is a strictly positive Lipschitz-smooth function. Let $x_j\in {\mathcal M}$, $j=1,2,\dots,N$ be $N$ independent random samples from distribution $\mu$. Also, let $\xi_j$, $j=1,2,\dots, N$ be independent random samples from a Gaussian random variable in ${\mathbb R}^n$ having covariance $\sigma^2I$, where $\sigma$ is less than a certain specified function of $d, V$ and $\tau$. We assume that we are given the data points $y_j=x_j+\xi_j,$ $j=1,2,\dots,N$, modelling random points of ${\mathcal M}$ with measurement noise. We develop an algorithm which produces from these data, with high probability, a $d$ dimensional submanifold ${\mathcal M}_o\subset {\mathbb R}^n$ whose Hausdorff distance to ${\mathcal M}$ is less than $Cd\sigma^2/\tau$ and whose reach is greater than $c{\tau}/d^6$ with universal constants $C,c > 0$. The number $N$ of random samples required depends almost linearly on $n$, polynomially on $\sigma^{-1}$ and exponentially on $d$.

The minimum energy path (MEP) describes the mechanism of reaction, and the energy barrier along the path can be used to calculate the reaction rate in thermal systems. The nudged elastic band (NEB) method is one of the most commonly used schemes to compute MEPs numerically. It approximates an MEP by a discrete set of configuration images, where the discretization size determines both computational cost and accuracy of the simulations. In this paper, we consider a discrete MEP to be a stationary state of the NEB method and prove an optimal convergence rate of the discrete MEP with respect to the number of images. Numerical simulations for the transitions of some several proto-typical model systems are performed to support the theory.

Holonomic functions play an essential role in Computer Algebra since they allow the application of many symbolic algorithms. Among all algorithmic attempts to find formulas for power series, the holonomic property remains the most important requirement to be satisfied by the function under consideration. The targeted functions mainly summarize that of meromorphic functions. However, expressions like $\tan(z)$, $z/(\exp(z)-1)$, $\sec(z)$, etc., particularly, reciprocals, quotients and compositions of holonomic functions, are generally not holonomic. Therefore their power series are inaccessible by the holonomic framework. From the mathematical dictionaries, one can observe that most of the known closed-form formulas of non-holonomic power series involve another sequence whose evaluation depends on some finite summations. In the case of $\tan(z)$ and $\sec(z)$ the corresponding sequences are the Bernoulli and Euler numbers, respectively. Thus providing a symbolic approach that yields complete representations when linear summations for power series coefficients of non-holonomic functions appear, might be seen as a step forward towards the representation of non-holonomic power series. By adapting the method of ansatz with undetermined coefficients, we build an algorithm that computes least-order quadratic differential equations with polynomial coefficients for a large class of non-holonomic functions. A differential equation resulting from this procedure is converted into a recurrence equation by applying the Cauchy product formula and rewriting powers into polynomials and derivatives into shifts. Finally, using enough initial values we are able to give normal form representations to characterize several non-holonomic power series and prove non-trivial identities. We discuss this algorithm and its implementation for Maple 2022.

We recall some of the history of the information-theoretic approach to deriving core results in probability theory and indicate parts of the recent resurgence of interest in this area with current progress along several interesting directions. Then we give a new information-theoretic proof of a finite version of de Finetti's classical representation theorem for finite-valued random variables. We derive an upper bound on the relative entropy between the distribution of the first $k$ in a sequence of $n$ exchangeable random variables, and an appropriate mixture over product distributions. The mixing measure is characterised as the law of the empirical measure of the original sequence, and de Finetti's result is recovered as a corollary. The proof is nicely motivated by the Gibbs conditioning principle in connection with statistical mechanics, and it follows along an appealing sequence of steps. The technical estimates required for these steps are obtained via the use of a collection of combinatorial tools known within information theory as `the method of types.'

The rapid recent progress in machine learning (ML) has raised a number of scientific questions that challenge the longstanding dogma of the field. One of the most important riddles is the good empirical generalization of overparameterized models. Overparameterized models are excessively complex with respect to the size of the training dataset, which results in them perfectly fitting (i.e., interpolating) the training data, which is usually noisy. Such interpolation of noisy data is traditionally associated with detrimental overfitting, and yet a wide range of interpolating models -- from simple linear models to deep neural networks -- have recently been observed to generalize extremely well on fresh test data. Indeed, the recently discovered double descent phenomenon has revealed that highly overparameterized models often improve over the best underparameterized model in test performance. Understanding learning in this overparameterized regime requires new theory and foundational empirical studies, even for the simplest case of the linear model. The underpinnings of this understanding have been laid in very recent analyses of overparameterized linear regression and related statistical learning tasks, which resulted in precise analytic characterizations of double descent. This paper provides a succinct overview of this emerging theory of overparameterized ML (henceforth abbreviated as TOPML) that explains these recent findings through a statistical signal processing perspective. We emphasize the unique aspects that define the TOPML research area as a subfield of modern ML theory and outline interesting open questions that remain.

This paper focuses on the expected difference in borrower's repayment when there is a change in the lender's credit decisions. Classical estimators overlook the confounding effects and hence the estimation error can be magnificent. As such, we propose another approach to construct the estimators such that the error can be greatly reduced. The proposed estimators are shown to be unbiased, consistent, and robust through a combination of theoretical analysis and numerical testing. Moreover, we compare the power of estimating the causal quantities between the classical estimators and the proposed estimators. The comparison is tested across a wide range of models, including linear regression models, tree-based models, and neural network-based models, under different simulated datasets that exhibit different levels of causality, different degrees of nonlinearity, and different distributional properties. Most importantly, we apply our approaches to a large observational dataset provided by a global technology firm that operates in both the e-commerce and the lending business. We find that the relative reduction of estimation error is strikingly substantial if the causal effects are accounted for correctly.