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Approximating functions by a linear span of truncated basis sets is a standard procedure for the numerical solution of differential and integral equations. Commonly used concepts of approximation methods are well-posed and convergent, by…

This paper proposes a new methodology in linear time-periodic (LTP) system identification. In contrast to previous methods that totally separate dynamics at different tag times for identification, the method focuses on imposing appropriate…

Sequential estimation of the success probability $p$ in inverse binomial sampling is considered in this paper. For any estimator $\hat p$, its quality is measured by the risk associated with normalized loss functions of linear-linear or…

Models are consistently treated as approximations and all procedures are consistent with this. They do not treat the model as being true. In this context $p$-values are one measure of approximation, a small $p$-value indicating a poor…

Let $R_{1,2}$ be scalar Riesz transforms on $\mathbb{R}^2$. We prove that the $L^p$ norms of $k$-th powers of the operator $R_2+iR_1$ behave exactly as $|k|^{1-2/p}p$, uniformly in $k\in\mathbb{Z}\backslash\{0\}$, $p\geq2$. This gives a…

We show that the natural scaling of measurement for a particular problem defines the most likely probability distribution of observations taken from that measurement scale. Our approach extends the method of maximum entropy to use…

The standard analytic solution to the DGLAP equation in Mellin space is improved by resumming the large x divergences. Explicit results are given to next-to-leading order and next-to-leading logarithmic accuracy. Numerically, the…

Fine-tuning LLMs for classification typically maps inputs directly to labels. We ask whether attaching brief explanations to each label during fine-tuning yields better models. We evaluate conversational response quality along three axes:…

The latest research on Large Language Models (LLMs) has demonstrated significant advancement in the field of Natural Language Processing (NLP). However, despite this progress, there is still a lack of reliability in these models. This is…

We give a characterization of $L^{p}(\sigma)$ for uniformly rectifiable measures $\sigma$ using Tolsa's $\alpha$-numbers, by showing, for $1<p<\infty$ and $f\in L^{p}(\sigma)$, that \[ \lVert f\rVert_{L^{p}(\sigma)}\sim…

Generating semantically coherent text requires a robust internal representation of linguistic structures, which traditional embedding techniques often fail to capture adequately. A novel approach, Latent Lexical Projection (LLP), is…

Histograms, i.e., piece-wise constant approximations, are a popular tool used to represent data distributions. Traditionally, the difference between the histogram and the underlying distribution (i.e., the approximation error) is measured…

The suitable basis functions for approximating periodic function are periodic, trigonometric functions. When the function is not periodic, a viable alternative is to consider polynomials as basis functions. In this paper we will point out…

The first order loss function and its complementary function are extensively used in practical settings. When the random variable of interest is normally distributed, the first order loss function can be easily expressed in terms of the…

We fit the exponent of the Pareto distribution, that is equivalent or can approximate the continuous power law distribution given a cutoff point, using linear regression (LR). We use LR on the logged variables of the empirical tail (one…

Let $m$ and $k \geq 2$ be positive integers. We show that polynomial $P = (1+x)^m(1+x^k)$ is strongly unimodal (frequently known as {\it log concave\/}) if and only if $m \geq k^2 -3$; this is also the criterion for $P$ to be merely…

Logistic-regression calibration and fusion are potential steps in the calculation of forensic likelihood ratios. The present paper provides a tutorial on logistic-regression calibration and fusion at a practical conceptual level with…

Recent works in dimensionality reduction for regression tasks have introduced the notion of sensitivity, an estimate of the importance of a specific datapoint in a dataset, offering provable guarantees on the quality of the approximation…

This paper presents a simple notion of proof net for multiplicative linear logic with units. Cut elimination is direct and strongly normalising, in contrast to previous approaches which resorted to moving jumps (attachments) of par units…

The sequence of 1/2-discrepancy sums of $\{x + i \theta \bmod 1\}$ is realized through a sequence of substitutions on an alphabet of three symbols; particular attention is paid to $x=0$. The first application is to show that any asymptotic…