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This study uses a pointwise statistical approach to analyze Near-Infrared Spectroscopy (NIRS) signals in preterm infants with and without Patent Ductus Arteriosus (PDA). The analysis focuses on three signals: blood oxygenation (SpO2), cerebral oxygenation (rSO2-1), and renal oxygenation (rSO2-2), across three newborn groups: without PDA (no-PDA), with hemodynamically insignificant PDA (PDA), and with hemodynamically significant PDA (hsPDA). While NIRS is widely used in medicine, its research, featuring statistical analysis, has been limited. Smoothed signals were tested using pointwise ANOVA and Tukey HSD to detect significant group differences. Results showed distinct patterns in rSO2-1 and rSO2-2, with the hsPDA group standing out in rSO2-1 and the no-PDA group in rSO2-2, demonstrating the value of this method in biomedical signal analysis. Pointwise ANOVA shows more time periods with significant differences compared to the SpO2 signal. The time period with the most significant differences is between 2 and 6 h, with additional peaks of p-values below 0.05 occurring before 2 h. These findings demonstrate the value of FDA in improving statistical analysis of biomedical NIRS signals and support its use in future research.

We investigate the pointwise convergence of the solution to the fractional Schrödinger equation in ℝ². By establishing Hs (ℝ²) L³(ℝ²) estimates for the associated maximal operator provided that s > 1/3, we improve the previous result obtained by Miao, Yang, and Zheng [19]. Our estimates extend the refined Strichartz estimates obtained by Du, Guth, and Li [10] to a general class of elliptic functions.

In this paper, as a generalization of pointwise slant submanifolds [B-Y. Chen and O. J. Garay, Pointwise slant submanifolds in almost Hermitian manifolds, Turk J Math 36, (2012), 630-640.], pointwise slant submersions [J.W.Lee and B. Ṣahin, Pointwise slant submersions, Bulletin of the Korean Mathematical Sosiety, 51(4), (2014), 115-1126.] and pointwise slant Riemannian maps [Y. Gündüzalp and M. A. Akyol, Pointwise slant Riemannian maps from Kaehler manifolds, Journal of Geometry and Physics, 179, (2002), 104589.], we introduce pointwise semi-slant Riemannian maps (briefly, PSSR maps) from almost Hermitian manifolds to Riemannian manifolds, present examples and characterizations. We also investigate the harmonicity of such maps. Moreover, we give Chen-Ricci inequality for a PSSR map. Finally, we study some curvature relations in complex space forms, involving Casorati curvatures for PSSR maps.

An initial-boundary value problem with a Caputo time derivative of fractional order α ϵ (0, 1) is considered, solutions of which typically exhibit a singular behavior at an initial time. For this problem, we give a simple and general numerical-stability analysis using barrier functions, which yields sharp pointwise-in-time error bounds on quasi-graded temporal meshes with arbitrary degree of grading. L1-type and Alikhanov-type discretization in time are considered. In particular, those results imply that milder (compared to the optimal) grading yields optimal convergence rates in positive time. Semidiscretizations in time and full discretizations are addressed. The theoretical findings are illustrated by numerical experiments.

We introduce a pointwise variant of the Assouad dimension for measures on metric spaces, and study its properties in relation to the global Assouad dimension. We show that, in general, the value of the pointwise Assouad dimension may differ from the global counterpart, but in many classical cases, the pointwise Assouad dimension exhibits similar exact dimensionality properties as the classical local dimension, namely it equals the global Assouad dimension almost everywhere. We also prove an explicit formula for the Assouad dimension of certain invariant measures with place-dependent probabilities supported on self-conformal sets.

In this paper, we introduce the notion of pointwise hemi-slant sub-manifolds of nearly Kaehler manifolds. Further, we study their warped products and prove the necessary and sufficient condition that a point-wise hemi-slant submanifold to be a warped product manifold. Also, we establish a sharp inequality for the pointwise hemi-slant warped product submanifolds of the form M = M⊥ ×f Mθ which is mixed totally geodesic in an arbitrary nearly Kaehler manifold ~M. The equality case is also discussed.

We study the rates of estimation of finite mixing distributions, that is, the parameters of the mixture. We prove that under some regularity and strong identifiability conditions, around a given mixing distribution with m₀ components, the optimal local minimax rate of estimation of a mixing distribution with m components is n −1/(4(m−m₀)+2). This corrects a previous paper by Chen [Ann. Statist. 23 (1995) 221–233]. By contrast, it turns out that there are estimators with a (nonuniform) pointwise rate of estimation of n −1/2 for all mixing distributions with a finite number of components.

The problem of how to properly quantify redundant information is an open question that has been the subject of much recent research. Redundant information refers to information about a target variable S that is common to two or more predictor variables X i . It can be thought of as quantifying overlapping information content or similarities in the representation of S between the X i . We present a new measure of redundancy which measures the common change in surprisal shared between variables at the local or pointwise level. We provide a game-theoretic operational definition of unique information, and use this to derive constraints which are used to obtain a maximum entropy distribution. Redundancy is then calculated from this maximum entropy distribution by counting only those local co-information terms which admit an unambiguous interpretation as redundant information. We show how this redundancy measure can be used within the framework of the Partial Information Decomposition (PID) to give an intuitive decomposition of the multivariate mutual information into redundant, unique and synergistic contributions. We compare our new measure to existing approaches over a range of example systems, including continuous Gaussian variables. Matlab code for the measure is provided, including all considered examples.

The solution to the system of equations of the descriptor linear discrete-time with different fractional orders is derived by the use of the Drazin inverse of matrices. This solution is applied to analysis of the pointwise completeness and the pointwise degeneracy of the descriptor discrete-time linear systems with different fractional orders. Necessary and sufficient conditions for the pointwise completeness and the pointwise degeneracy of the descriptor discrete–time linear systems with different fractional orders are established. The proposed methods are illustrated by numerical examples.

What are the distinct ways in which a set of predictor variables can provide information about a target variable? When does a variable provide unique information, when do variables share redundant information, and when do variables combine synergistically to provide complementary information? The redundancy lattice from the partial information decomposition of Williams and Beer provided a promising glimpse at the answer to these questions. However, this structure was constructed using a much criticised measure of redundant information, and despite sustained research, no completely satisfactory replacement measure has been proposed. In this paper, we take a different approach, applying the axiomatic derivation of the redundancy lattice to a single realisation from a set of discrete variables. To overcome the difficulty associated with signed pointwise mutual information, we apply this decomposition separately to the unsigned entropic components of pointwise mutual information which we refer to as the specificity and ambiguity. This yields a separate redundancy lattice for each component. Then based upon an operational interpretation of redundancy, we define measures of redundant specificity and ambiguity enabling us to evaluate the partial information atoms in each lattice. These atoms can be recombined to yield the sought-after multivariate information decomposition. We apply this framework to canonical examples from the literature and discuss the results and the various properties of the decomposition. In particular, the pointwise decomposition using specificity and ambiguity satisfies a chain rule over target variables, which provides new insights into the so-called two-bit-copy example.