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Background: Using neuroimaging techniques, growing evidence has suggested that the choroid plexus (CP) volume is enlarged in multiple neurodegenerative diseases, including amyotrophic lateral sclerosis (ALS). Notably, the CP has been suggested to play an important role in inflammation-induced CNS damage under disease conditions. However, to our knowledge, no study has investigated the relationships between peripheral inflammation and CP volume in sporadic ALS patients. Thus, in this study, we aimed to verify CP enlargement and explore its association with peripheral inflammation in vivo in independent ALS cohorts. Methods: Based on structural MRI data, CP volume was measured using Gaussian mixture models and further manually corrected in two independent cohorts of sporadic ALS patients and healthy controls (HCs). Serum inflammatory protein levels were measured using a novel high-sensitivity Olink proximity extension assay (PEA) technique. Xtreme gradient boosting (XGBoost) was used to explore the contribution of peripheral inflammatory factors to CP enlargement. Then, partial correlation analyses were performed. Results: CP volumes were significantly higher in ALS patients than in HCs in the independent cohorts. Compared with HCs, serum levels of CRP, IL-6, CXCL10, and 35 other inflammatory factors were significantly increased in ALS patients. Using the XGBoost approach, we established a model-based importance of features, and the top three predictors of CP volume in ALS patients were CRP, IL-6, and CXCL10 (with gains of 0.24, 0.18, and 0.15, respectively). Correlation analyses revealed that CRP, IL-6, and CXCL10 were significantly associated with CP volume in ALS patients (r = 0.462 ∼ 0.636, p < 0.001). Conclusion: Our study is the first to reveal a consistent and replicable contribution of peripheral inflammation to CP enlargement in vivo in sporadic ALS patients. Given that CP enlargement has been recently detected in other brain diseases, these findings should consider extending to other disease conditions with a peripheral inflammatory component. [ABSTRACT FROM AUTHOR]

Contraction properties of transport maps between probability measures play an important role in the theory of functional inequalities. The actual construction of such maps, however, is a non-trivial task and, so far, relies mostly on the theory of optimal transport. In this work, we take advantage of the infinite-dimensional nature of the Gaussian measure and construct a new transport map, based on the Föllmer process, which pushes forward the Wiener measure onto probability measures on Euclidean spaces. Utilizing the tools of the Malliavin and stochastic calculus in Wiener space, we show that this Brownian transport map is a contraction in various settings where the analogous questions for optimal transport maps are open. The contraction properties of the Brownian transport map enable us to prove functional inequalities in Euclidean spaces, which are either completely new or improve on current results. Further and related applications of our contraction results are the existence of Stein kernels with desirable properties (which lead to new central limit theorems), as well as new insights into the Kannan–Lovász–Simonovits conjecture. We go beyond the Euclidean setting and address the problem of contractions on the Wiener space itself. We show that optimal transport maps and causal optimal transport maps (which are related to Brownian transport maps) between the Wiener measure and other target measures on Wiener space exhibit very different behaviors. [ABSTRACT FROM AUTHOR]

We prove that if q 1 , ⋯ , q m : R n → R are quadratic forms in variables x 1 , ⋯ , x n such that each q k depends on at most r variables and each q k has common variables with at most r other forms, then the average value of the product (1 + q 1) ⋯ (1 + q m) with respect to the standard Gaussian measure in R n can be approximated within relative error ϵ > 0 in quasi-polynomial n O (1) m O (ln m - ln ϵ) time, provided | q k (x) | ⩽ γ ‖ x ‖ 2 / r for some absolute constant γ > 0 and k = 1 , ... , m . The integral in question is viewed as the independence polynomial of an auxiliary weighted graph and then the method of polynomial interpolation is applied. When q k are interpreted as pairwise squared distances for configurations of points in Euclidean space, the average can be interpreted as the partition function of systems of particles with mollified logarithmic potentials. We sketch possible applications to testing the feasibility of systems of real quadratic equations and to computing permanents of positive definite Hermitian matrices. [ABSTRACT FROM AUTHOR]

In this note, we obtain a Gaussian concentration inequality for a class of non-Lipschitz functions. In the one-dimensional case, our results supplement those established by Paouris and Valettas [Ann. Probab. 46 (2018), pp. 1441–1454]. [ABSTRACT FROM AUTHOR]

The present article describes the precise structure of the L p -spaces of projective limit measures by introducing a category-theoretical perspective. This analysis is applied to measures on vector spaces and in particular to Gaussian measures on nuclear topological vector spaces. A simple application to constructive quantum field theory (QFT) is given through the Osterwalder–Schrader axioms. [ABSTRACT FROM AUTHOR]

Skin diseases and lesions can be ambiguous to recognize due to the similarity of lesions and enhanced imaging features. In this study, we compared three cutting-edge deep learning frameworks for dermoscopic segmentation: U-Net, SegAN, and MultiResUNet. We used a dermoscopic dataset including detailed lesion annotations with segmentation masks to help train and evaluate models on the precise localization of melanomas. SegAN is a special type of Generative Adversarial Network (GAN) that introduces a new architecture by adding generator and discriminator steps. U-Net has become a common strategy in segmentation to encode and decode image features for limited data. MultiResUNet is a U-Net-based architecture that overcomes the insufficient data problem in medical imaging by extracting contextual details. We trained the three frameworks on colored images after preprocessing. We added incremental Gaussian noise to measure the robustness of segmentation performance. We evaluated the frameworks using the following parameters: accuracy, sensitivity, specificity, Dice and Jaccard coefficients. Our accuracy results show that SegAN (92%) and MultiResUNet (92%) both outperform U-Net (86%), which is a well-known segmentation framework for skin lesion analysis. MultiResUNet sensitivity (96%) outperforms the methods in the challenge leaderboard. These results suggest that SegAN and MultiResUNet are more resistant techniques against noise in dermoscopic segmentation. [ABSTRACT FROM AUTHOR]

This paper deals with compatibility of space‐time kernels with (either) full, spatially dynamical, or space‐time compact support. We deal with the dilemma of statistical accuracy versus computational scalability, which are in a notorious trade‐off. Apparently, models with full support ensure maximal information but are computationally expensive, while compactly supported models achieve computational scalability at the expense of loss of information. Hence, an inspection of whether these models might be compatible is necessary. The criterion we use for such an inspection is based on equivalence of Gaussian measures. We provide sufficient conditions for space‐time compatibility. As a corollary, we deduce implications in terms of maximum likelihood estimation and misspecified kriging prediction under fixed domain asymptotics. Some results of independent interest relate about the space‐time spectrum associated with the classes of kernels proposed in the paper. [ABSTRACT FROM AUTHOR]

In general, raw image may suffer from various uncertain distortions arising due to many factors like low dynamic range of imaging device, low illumination, bad weather conditions. In this paper, a new idea of fuzzy peer group-based image denoising algorithm has been presented with a novel similarity measure function. Unlikely the Gaussian membership function based conventional similarity measure; here proposed method hired the local spatial information induced similarity measure together with Gaussian membership function by exploiting the significance of both. Using the proposed similarity measure, fuzzy peer group of similar pixels contrived and color image having contamination of mixed noise i.e., impulse noise and Gaussian noise is decontaminated with refashioned form of bilateral filter. The algorithm proposed has been compared with some state-of-the-art methods in studies and other techniques. The experimental performance shows that proposed approach is having leading filtering outcome with edge preserving and color information from existing methods. [ABSTRACT FROM AUTHOR]

ABSTRACT The ICH E9(R1) Addendum (International Council for Harmonization 2019) suggests treatment‐policy as one of several strategies for addressing intercurrent events such as treatment withdrawal when defining an estimand. This strategy requires the monitoring of patients and collection of primary outcome data following termination of randomised treatment. However, when patients withdraw from a study early before completion this creates true missing data complicating the analysis. One possible way forward uses multiple imputation to replace the missing data based on a model for outcome on‐ and off‐treatment prior to study withdrawal, often referred to as retrieved dropout multiple imputation. This article introduces a novel approach to parameterising this imputation model so that those parameters which may be difficult to estimate have mildly informative Bayesian priors applied during the imputation stage. A core reference‐based model is combined with a retrieved dropout compliance model, using both on‐ and off‐treatment data, to form an extended model for the purposes of imputation. This alleviates the problem of specifying a complex set of analysis rules to accommodate situations where parameters which influence the estimated value are not estimable, or are poorly estimated leading to unrealistically large standard errors in the resulting analysis. We refer to this new approach as retrieved dropout reference‐base centred multiple imputation. [ABSTRACT FROM AUTHOR]

Let |$(M^{n},g,f)$| be a Ricci shrinker such that |$\text{Ric}_{f}=\frac{1}{2}g$| and the measure induced by the weighted volume element |$(4\pi)^{-\frac{n}{2}}e^{-f}dv_{g}$| is a probability measure. Given a point |$p\in M$|⁠ , we consider two probability measures defined in the tangent space |$T_{p}M$|⁠ , namely the Gaussian measure |$\gamma $| and the measure |$\overline{\nu }$| induced by the exponential map of |$M$| to |$p$|⁠. In this paper, we prove a result that provides an upper estimate for the Wasserstein distance with respect to the Euclidean metric |$g_{0}$| between the measures |$\overline{\nu }$| and |$\gamma $|⁠ , and which also elucidates the rigidity implications resulting from this estimate. [ABSTRACT FROM AUTHOR]

Randomized quadratures for integrating functions in Sobolev spaces of order \alpha \ge 1, where the integrability condition is with respect to the Gaussian measure, are considered. In this function space, the optimal rate for the worst-case root-mean-squared error (RMSE) is established. Here, optimality is for a general class of quadratures, in which adaptive non-linear algorithms with a possibly varying number of function evaluations are also allowed. The optimal rate is given by showing matching bounds. First, a lower bound on the worst-case RMSE of O(n^{-\alpha -1/2}) is proven, where n denotes an upper bound on the expected number of function evaluations. It turns out that a suitably randomized trapezoidal rule attains this rate, up to a logarithmic factor. A practical error estimator for this trapezoidal rule is also presented. Numerical results support our theory. [ABSTRACT FROM AUTHOR]

We formulate a plausible conjecture for the optimal Ehrhard-type inequality for convex symmetric sets with respect to the Gaussian measure. Namely, letting J_{k-1}(s)=\int ^s_0 t^{k-1} e^{-\frac {t^2}{2}}dt and c_{k-1}=J_{k-1}(+\infty), we conjecture that the function F:[0,1]\rightarrow \mathbb {R}, given by \begin{equation*} F(a)= \sum _{k=1}^n 1_{a\in E_k}\cdot (\beta _k J_{k-1}^{-1}(c_{k-1} a)+\alpha _k) \end{equation*} (with an appropriate choice of a decomposition [0,1]=\cup _{i} E_i and coefficients \alpha _i, \beta _i) satisfies, for all symmetric convex sets K and L, and any \lambda \in [0,1], \begin{equation*} F\left (\gamma (\lambda K+(1-\lambda)L)\right)\geq \lambda F\left (\gamma (K)\right)+(1-\lambda) F\left (\gamma (L)\right). \end{equation*} We explain that this conjecture is "the most optimistic possible", and is equivalent to the fact that for any symmetric convex set K, its Gaussian concavity power p_s(K,\gamma) is greater than or equal to p_s(RB^k_2\times \mathbb {R}^{n-k},\gamma), for some k\in \{1,\dots,n\}. We call the sets RB^k_2\times \mathbb {R}^{n-k} round k-cylinders ; they also appear as the conjectured Gaussian isoperimetric minimizers for symmetric sets, see Heilman [Amer. J. Math. 143 (2021), pp. 53–94]. In this manuscript, we make progress towards this question, and show that for any symmetric convex set K in \mathbb {R}^n, \begin{equation*} p_s(K,\gamma)\geq \sup _{F\in L^2(K,\gamma)\cap Lip(K):\,\int F=1} \left (2T_{\gamma }^F(K)-Var(F)\right)+\frac {1}{n-\mathbb {E}X^2}, \end{equation*} where T_{\gamma }^F(K) is the F-torsional rigidity of K with respect to the Gaussian measure. Moreover, the equality holds if and only if K=RB^k_2\times \mathbb {R}^{n-k} for some R>0 and k=1,\dots,n. As a consequence, we get \begin{equation*} p_s(K,\gamma)\geq Q(\mathbb {E}|X|^2, \mathbb {E}\|X\|_K^4, \mathbb {E}\|X\|^2_K, r(K)), \end{equation*} where Q is a certain rational function of degree 2, the expectation is taken with respect to the restriction of the Gaussian measure onto K, \|\cdot \|_K is the Minkowski functional of K, and r(K) is the in-radius of K. The result follows via a combination of some novel estimates, the L2 method (previously studied by several authors, notably Kolesnikov and Milman [J. Geom. Anal. 27 (2017), pp. 1680–1702; Amer. J. Math. 140 (2018), pp. 1147–1185; Geometric aspects of functional analysis , Springer, Cham, 2017; Mem. Amer. Math. Soc. 277 (2022), v+78 pp.], Kolesnikov and the author [Adv. Math. 384 (2021), 23 pp.], Hosle, Kolesnikov, and the author [J. Geom. Anal. 31 (2021), pp. 5799–5836], Colesanti [Commun. Contemp. Math. 10 (2008), pp. 765–772], Colesanti, the author, and Marsiglietti [J. Funct. Anal. 273 (2017), pp. 1120–1139], Eskenazis and Moschidis [J. Funct. Anal. 280 (2021), 19 pp.]), and the analysis of the Gaussian torsional rigidity. As an auxiliary result on the way to the equality case characterization, we characterize the equality cases in the "convex set version" of the Brascamp-Lieb inequality, and moreover, obtain a quantitative stability version in the case of the standard Gaussian measure; this may be of independent interest. All the equality case characterizations rely on the careful analysis of the smooth case, the stability versions via trace theory, and local approximation arguments. In addition, we provide a non-sharp estimate for a function F whose composition with \gamma (K) is concave in the Minkowski sense for all symmetric convex sets. [ABSTRACT FROM AUTHOR]

Gaussian process regression (GPR) models have become popular as fast alternative models for complex computer codes. For complex computer code (CC) with multivariate outputs, a GPR model can be constructed separately for each CC output, ignoring the correlation between the different outputs. However, this may lead to poor performance of the GPR model. To tackle this problem, multivariate GPR models are used for complex multivariate deterministic computer codes. This paper proposes measures for quantifying uncertainty and checking the assumptions that are proposed in building multivariate GPR models. For comparison, we also constructed a univariate GPR model for each CC output to investigate the effect of ignoring the correlation between the different outputs. We found that the multivariate GPR model outperforms the univariate GPR model as it provides more accurate predictions and quantifies uncertainty about the CC outputs appropriately. [ABSTRACT FROM AUTHOR]

Background: The aim of the study was to investigate the muscle differences in children with osteogenesis imperfecta (OI) using opportunistic low-dose chest CT and to compare different methods for the segmentation of muscle in children. Methods: This single center retrospective study enrolled children with OI and controls undergoing opportunistic low-dose chest CT obtained during the COVID pandemic. From the CT images, muscle size (cross-sectional area) and density (mean Hounsfield Units [HU]) of the trunk muscles were measured at the mid-T4 and the mid-T10 level using two methods, the fixed thresholds and the Gaussian mixture model. The Bland-Altman method was also used to compute the strength of agreement between two methods. Comparison of muscle results between OI and controls were analyzed with Student t tests. Results: 20 children with OI (mean age, 9.1 ± 3.3 years, 15 males) and 40 age- and sex-matched controls were enrolled. Mean differences between two methods were good. Children with OI had lower T4 and T10 muscle density than controls measured by the fixed thresholds (41.2 HU vs. 48.0 HU, p < 0.01; 37.3 HU vs. 45.9 HU, p < 0.01). However, children with OI had lower T4 muscle size, T4 muscle density, T10 muscle size and T10 muscle density than controls measured by the Gaussian mixture model (110.9 vs. 127.2 cm2, p = 0.03; 44.6 HU vs. 51.3 HU, p < 0.01; 72.6 vs. 88.0 cm2, p = 0.01; 41.6 HU vs. 50.3 HU, p < 0.01, respectively). Conclusions: Children with OI had lower trunk muscle density indicating that OI might also impair muscle quality. Moreover, the fixed thresholds may not be suitable for segmentation of muscle in children. [ABSTRACT FROM AUTHOR]

We prove the invariance of the Gibbs measure under the dynamics of the three-dimensional cubic wave equation, which is also known as the hyperbolic Φ 3 4 -model. This result is the hyperbolic counterpart to seminal works on the parabolic Φ 3 4 -model by Hairer (Invent. Math. 198(2):269–504, 2014) and Hairer-Matetski (Ann. Probab. 46(3):1651–1709, 2018). The heart of the matter lies in establishing local in time existence and uniqueness of solutions on the statistical ensemble, which is achieved by using a para-controlled ansatz for the solution, the analytical framework of the random tensor theory, and the combinatorial molecule estimates. The singularity of the Gibbs measure with respect to the Gaussian free field brings out a new caloric representation of the Gibbs measure and a synergy between the parabolic and hyperbolic theories embodied in the analysis of heat-wave stochastic objects. Furthermore from a purely hyperbolic standpoint our argument relies on key new ingredients that include a hidden cancellation between sextic stochastic objects and a new bilinear random tensor estimate. [ABSTRACT FROM AUTHOR]

This paper examines the tractability of multivariate approximation problems under the normalized error criterion for a zero-mean Gaussian measure in an average-case setting. The Gaussian measure is associated with a covariance kernel, which is represented by the tensor product of one-dimensional kernels corresponding to Euler and Wiener integrated processes with non-negative and nondecreasing smoothness parameters { r d } d ∈ N . We give matching sufficient and necessary conditions for various concepts of tractability in terms of the asymptotic properties of the regularity parameters, except for (s, 0)-WT. [ABSTRACT FROM AUTHOR]

The symplectic eigenvalues play a significant role in the finite-mode quantum information theory, and Williamson's normal form proves to be a valuable tool in this area. Understanding the symplectic spectrum of a Gaussian Covariance Operator is a crucial task. Recently, in 2018, an infinite-dimensional analogue of Williamson's Normal form was discovered, which has been instrumental in studying the infinite-mode Gaussian quantum states. However, most existing results pertain to finite-dimensional operators, leaving a dearth of literature in the infinite-dimensional context. The focus of this article is on employing approximation techniques to estimate the symplectic spectrum of certain infinite-dimensional operators. These techniques are well-suited for a particular class of operators, including specific types of infinite-mode Gaussian Covariance Operators. Our approach involves computing the Williamson's Normal form and deriving bounds for the symplectic spectrum of these operators. As a practical application, we explicitly compute the symplectic spectrum of Gaussian Covariance Operators. Through this research, we aim to contribute to the understanding of symplectic eigenvalues in the context of infinite-dimensional operators, opening new avenues for exploration in quantum information theory and related fields. [ABSTRACT FROM AUTHOR]

This paper is devoted to introducing and investigating the bounded variation capacity and the perimeter in the abstract Wiener space X, thereby discovering some related inequalities. Functions of bounded variation in an abstract Wiener space X have been studied by many scholars. As the continuation of this research, we define the corresponding BV capacity cap H ⁡ (⋅) (now called abstract Wiener BV capacity) and investigate its properties. We also investigate some properties of sets of finite γ-perimeter, with γ being a Gaussian measure. Subsequently, the isocapacitary inequality associated with cap H ⁡ (⋅) is presented and we are able to show that it is equivalent to the Gaussian isoperimetric inequality. Finally, we prove that every set of finite γ-perimeter in X has mean curvature in L 1 ⁢ (X , γ) . [ABSTRACT FROM AUTHOR]

Given an ample real Hermitian holomorphic line bundle L over a real algebraic variety X, the space of real holomorphic sections of Lxd inherits a natural Gaussian probability measure. We prove that the probability that the zero locus of a real holomorphic section s of Lxd defines a maximal hypersurface tends to 0 exponentially fast as d goes to infinity. This extends to any dimension a result of Gayet and Welschinger (2011) valid for maximal real algebraic curves inside a real algebraic surface. The starting point is a low degree approximation property which relates the topology of the real vanishing locus of a real holomorphic section of Lxd with the topology of the real vanishing locus a real holomorphic section of Lxd0 for a sufficiently smaller d0 < d. Such a statement is inspired by the recent work of Diatta and Lerario (2022). [ABSTRACT FROM AUTHOR]

A simple condition is given that is sufficient to determine whether a measure that is absolutely continuous with respect to a Gaußian measure on the space of distributions is reflection positive. It readily generalises conventional lattice results to an abstract setting, enabling the construction of many reflection positive measures that are not supported on lattices. [ABSTRACT FROM AUTHOR]

We investigate the transient dynamics of the Fisher equation under nonlinear diffusion and fractional operators. Firstly, we investigate the effects of the nonlinear diffusivity parameter in the integer-order Fisher equation, by considering a Gaussian distribution as the initial condition. Measuring the spread of the Gaussian distribution by u (0 , t) − 2 , our results show that the solution reaches a steady state governed by the parameters present in the logistic function in Fisher's equation. The initial transient is an anomalous diffusion process, but a power law cannot describe the whole transient. In this sense, the main novelty of this work is to show that a q-exponential function gives a better description of the transient dynamics. In addition to this result, we extend the Fisher equation via non-integer operators. As a fractional definition, we employ the Caputo fractional derivative and use a discretized system for the numerical approach according to finite difference schemes. We consider the numerical solutions in three scenarios: fractional differential operators acting in time, space, and in both variables. Our results show that the time to reach the steady solution strongly depends on the fractional order of the differential operator, with more influence by the time operator. Our main finding shows that a generalized q-exponential, present in the Tsallis formalism, describes the transient dynamics. The adjustment parameters of the q-exponential depend on the fractional order, connecting the generalized thermostatistics with the anomalous relaxation promoted by the fractional operators in time and space. [ABSTRACT FROM AUTHOR]

In this paper, we study the dynamics of an operator \mathcal T naturally associated to the so-called Collatz map , which maps an integer n \geq 0 to n / 2 if n is even and 3n + 1 if n is odd. This operator \mathcal T is defined on certain weighted Bergman spaces \mathcal B ^2 _\omega of analytic functions on the unit disk. Building on previous work of Neklyudov, we show that \mathcal T is hypercyclic on \mathcal B ^2 _\omega, independently of whether the Collatz Conjecture holds true or not. Under some assumptions on the weight \omega, we show that \mathcal T is actually ergodic with respect to a Gaussian measure with full support, and thus frequently hypercyclic and chaotic. [ABSTRACT FROM AUTHOR]

We investigate the numerical approximation of integrals over |$\mathbb{R}^{d}$| equipped with the standard Gaussian measure |$\gamma $| for integrands belonging to the Gaussian-weighted Sobolev spaces |$W^{\alpha }_{p}(\mathbb{R}^{d}, \gamma)$| of mixed smoothness |$\alpha \in \mathbb{N}$| for |$1 < p < \infty $|⁠. We prove the asymptotic order of the convergence of optimal quadratures based on |$n$| integration nodes and propose a novel method for constructing asymptotically optimal quadratures. As for related problems, we establish by a similar technique the asymptotic order of the linear, Kolmogorov and sampling |$n$| -widths in the Gaussian-weighted space |$L_{q}(\mathbb{R}^{d}, \gamma)$| of the unit ball of |$W^{\alpha }_{p}(\mathbb{R}^{d}, \gamma)$| for |$1 \leq q < p < \infty $| and |$q=p=2$|⁠. [ABSTRACT FROM AUTHOR]

We study the phase reconstruction of signals f belonging to complex Gaussian shift-invariant spaces V ∞ (φ) from spectrogram measurements | G f (X) | where G is the Gabor transform and X ⊆ R 2 . An explicit reconstruction formula will demonstrate that such signals can be recovered from measurements located on parallel lines in the time-frequency plane by means of a Riesz basis expansion. Moreover, connectedness assumptions on |f| result in stability estimates in the situation where one aims to reconstruct f on compacts intervals. Driven by a recent observation that signals in Gaussian shift-invariant spaces are determined by lattice measurements (Grohs and Liehr in Injectivity of Gabor phase retrieval from lattice measurements. Appl. Comput. Harmon. Anal. 62, 173–193 (2023)) we prove a sampling result on the stable approximation from finitely many spectrogram samples. The resulting algorithm provides a provably stable and convergent approximation technique. In addition, it constitutes a method of approximating signals in function spaces beyond V ∞ (φ) , such as Paley–Wiener spaces. [ABSTRACT FROM AUTHOR]

We consider the Cauchy problem for the Hamiltonian system consisting of the Klein–Gordon field and an infinite harmonic crystal. The dynamics of the coupled system is translation-invariant with respect to the discrete subgroup of . The initial date is assumed to be a random function that is close to two spatially homogeneous (with respect to the subgroup ) processes when with some . We study the distribution of the solution at time and prove the weak convergence of to a Gaussian measure as . Moreover, we prove the convergence of the correlation functions to a limit and derive the explicit formulas for the covariance of the limit measure . We give an application to Gibbs measures. [ABSTRACT FROM AUTHOR]

Traffic time series anomaly detection has been intensively studied for years because of its potential applications in intelligent transportation. However, classical traffic anomaly detection methods often overlook the evolving dynamic associations between road network nodes, which leads to challenges in capturing the long-term temporal correlations, spatial characteristics, and abnormal node behaviors in datasets with high periodicity and trends, such as morning peak travel periods. In this paper, we propose a mirror temporal graph autoencoder (MTGAE) framework to explore anomalies and capture unseen nodes and the spatiotemporal correlation between nodes in the traffic network. Specifically, we propose the mirror temporal convolutional module to enhance feature extraction capabilities and capture hidden node-to-node features in the traffic network. Morever, we propose the graph convolutional gate recurrent unit cell (GCGRU CELL) module. This module uses Gaussian kernel functions to map data into a high-dimensional space, and enables the identification of anomalous information and potential anomalies within the complex interdependencies of the traffic network, based on prior knowledge and input data. We compared our work with several other advanced deep-learning anomaly detection models. Experimental results on the NYC dataset illustrate that our model works best compared to other models for traffic anomaly detection. [ABSTRACT FROM AUTHOR]

In this paper, we study the Onsager-Machlup function and its relationship to the Freidlin-Wentzell function for measures equivalent to arbitrary infinite dimensional Gaussian measures. The Onsager-Machlup function can serve as a density on infinite dimensional spaces, where a uniform measure does not exist, and has been seen as the Lagrangian for the "most likely element". The Freidlin-Wentzell rate function is the large deviations rate function for small-noise limits and has also been identified as a Lagrangian for the "most likely element". This leads to a conundrum - what is the relationship between these two functions? We show both pointwise and G-convergence (which is essentially the convergence of minimizers) of the Onsager-Machlup function under the small-noise limit to the Freidlin-Wentzell function - and give an expression for both. That is, we show that the small-noise limit of the most likely element is the most likely element in the small noise limit for infinite dimensional measures that are equivalent to a Gaussian. Examples of measures include the law of solutions to path-dependent stochastic differential equations and the law of an infinite system of random algebraic equations. [ABSTRACT FROM AUTHOR]

The paper discusses the effect of autocorrelations on process capability indices (PCIs).Confidence intervals are constructed for PCIs when data are autocorrelated. Approximate lower confidence limits for various Cpk are computed for AR(1) model. Industrial examples are considered to illustrate the results. [ABSTRACT FROM AUTHOR]

A longstanding paradox was first reported by David Miller in 1975 and highlighted by Karl Popper in 1979. Miller showed that the ranking of predictions from two theories, in terms of closeness to observation, appears to be reversed when the problem is transformed into a different mathematical space. He concluded that "... no false theory can ... be closer to the truth than is another theory". This flies in the face of normal scientific practice and is thus paradoxical; it is named here the "Miller-Popper paradox". This paper proposes a resolution of the paradox, through consideration of the inevitable errors and uncertainties in both observations and predictions. It is proved that, for linear transformations and Gaussian error distributions, the transformation between spaces creates no change in quantitative measures of "closeness-to-observation" when these measures are based in probability theory. The extension of this result to nonlinear transformations and to non-Gaussian error distributions is also discussed. These results demonstrate that concepts used in comparison of predictions with observations—concepts of "closeness", "consistency", "agreement", "falsification", etc.—all imply some knowledge of the uncertainty characteristics of both predictions and observations. [ABSTRACT FROM AUTHOR]

To reduce systematic uncertainties in Type Ia supernova (SN Ia) cosmology, the host galaxy dust law shape parameter, RV , must be accurately constrained. We thus develop a computationally inexpensive pipeline, Bird-Snack , to rapidly infer dust population distributions from optical-near-infrared SN colours at peak brightness, and determine which analysis choices significantly impact the population mean RV inference, |$\mu _{R_V}$|⁠. Our pipeline uses a 2D Gaussian process to measure peak BVriJH apparent magnitudes from SN light curves, and a hierarchical Bayesian model to simultaneously constrain population distributions of intrinsic and dust components. Fitting a low-to-moderate-reddening sample of 65 low-redshift SNe yields |$\mu _{R_V}=2.61^{+0.38}_{-0.35}$|⁠ , with |$68~{{\ \rm per\ cent}}(95~{{\ \rm per\ cent}})$| posterior upper bounds on the population dispersion, |$\sigma _{R_V}\lt 0.92(1.96)$|⁠. This result is robust to various analysis choices, including: the model for intrinsic colour variations, fitting the shape hyperparameter of a gamma dust extinction distribution, and cutting the sample based on the availability of data near peak. However, these choices may be important if statistical uncertainties are reduced. With larger near-future optical and near-infrared SN samples, Bird-Snack can be used to better constrain dust distributions, and investigate potential correlations with host galaxy properties. Bird-Snack is publicly available; the modular infrastructure facilitates rapid exploration of custom analysis choices, and quick fits to simulated data sets, for better interpretation of real-data inferences. [ABSTRACT FROM AUTHOR]

Real‐time estimating harmonic distortion levels near the Point of Interconnections of grid‐connected inverter‐based resources (IBRs) is critical for maintaining the operational reliability and security of the power grid, particularly during transients when IBRs generate excessive harmonics. Traditional methods like Total Harmonic Distortion (THD) are unsuitable for time‐varying waveforms as they suffer from aliasing, spectral leakage, and picket‐fence effects. This paper proposes a novel approach that utilizes convolution‐based measures for estimating harmonic distortion, which offers a more suitable way for ongoing real‐time applications. The first measure uses a 60 Hz sinusoidal function as the convolution kernel, while an enhanced measure employs the Gaussian function as the kernel. Two convolution‐based measures are proposed and compared with THD. The comparison criteria are based on the two significant features describing the variation of harmonics: min/max values and variation curvature of harmonics showing the accurate performance of proposed measures. Moreover, short‐term estimations of harmonic distortion are used to validate the effectiveness of the proposed measures, while long‐term estimations are used to determine the ability of the proposed measures to determine harmonic distortion contributions among power network components. The proposed technique's intrinsic characteristics provide a sampling window with low sensitivity to deviations in the fundamental frequency and signal's stationary condition to prevent aliasing and spectral leakage and rule out the impact of the picket‐fence effect on the harmonic distortion level estimation. [ABSTRACT FROM AUTHOR]

While global navigation satellite system (GNSS) technologies have always been the go-to solution for localization problems, they may not be the best choice for some Internet-of-Things (IoT) applications due to the incurred power consumption and cost. In this paper, we present an alternative satellite-based localization method exploiting the signature of Doppler shifts and angle-of-arrival measurements as seen by a low-Earth-orbit (LEO) satellite. We first derive the joint likelihood function of the measurements, which is represented as a combination of three Gaussian distributions. Then, we show that the maximum likelihood problem reduces to a more-efficient mean squared error minimization in the Gaussian case as inferred from real measurements we collected from low-Earth-orbit satellite using a tracking ground station. Thus, we propose utilizing a stochastic optimizer to search for the global minimum of the mean squared error, which represents the location of the ground IoT device as estimated by the satellite platform. The emulated results show that the IoT device localization, in such a realistic model, can be performed with sufficient accuracy for IoT applications. [ABSTRACT FROM AUTHOR]

The Bayesian inference is widely used in many scientific and engineering problems, especially in the linear inverse problems in infinite-dimensional setting where the unknowns are functions. In such problems, choosing an appropriate prior distribution is an important task. Especially when the function to infer has much detail information, such as many sharp jumps, corners, and the discontinuous and nonsmooth oscillation, the so-called total variation-Gaussian (TG) prior is proposed in function space to address it. However, the TG prior is easy to lead the blocky (staircase) effect in numerical results. In this work, we present a fractional order-TG (FTG) hybrid prior to deal with such problems, where the fractional order total variation (FTV) term is used to capture the detail information of the unknowns and simultaneously uses the Gaussian measure to ensure that it results in a well-defined posterior measure. For the numerical implementations of linear inverse problems in function spaces, we also propose an efficient independence sampler based on a transport map, which uses a proposal distribution derived from a diagonal map, and the acceptance probability associated to the proposal is independent of discretization dimensionality. And in order to take full advantage of the transport map, the hierarchical Bayesian framework is applied to flexibly determine the regularization parameter. Finally we provide some numerical examples to demonstrate the performance of the FTG prior and the efficiency and robustness of the proposed independence sampler method. [ABSTRACT FROM AUTHOR]

The Brunn-Minkowski Theory has seen several generalizations over the past century. Many of the core ideas have been generalized to measures. With the goal of framing these generalizations as a weighted Brunn-Minkowski theory, we prove the Minkowski existence theorem for a large class of Borel measures with continuous density, denoted by \Lambda ^n: for \nu a finite, even Borel measure on the unit sphere and even \mu \in \Lambda ^n, there exists a symmetric convex body K such that \begin{equation*} d\nu (u)=c_{\mu,K}dS^{\mu }_{K}(u), \end{equation*} where c_{\mu,K} is a quantity that depends on \mu and K and dS^{\mu }_{K}(u) is the surface area-measure of K with respect to \mu. Examples of measures in \Lambda ^n are homogeneous measures (with c_{\mu,K}=1) and probability measures with radially decreasing densities (e.g. the Gaussian measure). We will also consider weighted projection bodies \Pi _\mu K by classifying them and studying the isomorphic Shephard problem: if \mu and \nu are even, homogeneous measures with density and K and L are symmetric convex bodies such that \Pi _{\mu } K \subset \Pi _{\nu } L, then can one find an optimal quantity \mathcal {A}>0 such that \mu (K)\leq \mathcal {A}\nu (L)? Among other things, we show that, in the case where \mu =\nu and L is a projection body, \mathcal {A}=1. [ABSTRACT FROM AUTHOR]

Matched-field processing (MFP) for underwater source localization serves as a generalized beamforming approach that assesses the correlation between the received array data and a dictionary of replica vectors. In this study, the processing scheme of MFP is reformulated by computing a statistical metric between two Gaussian probability measures with the cross-spectral density matrices (CSDMs). To achieve this, the Wasserstein metric, a widely used notion of metric in the space of probability measures, is employed for developing the processor to attach the intrinsic properties of CSDMs, expressing the underlying optimal value of the statistic. The Wasserstein processor uses the embedded metric structure to suppress ambiguities, resulting in the ability to distinguish between multiple sources. In this foundation, a multifrequency processor that combines the information at different frequencies is derived, providing improved localization statistics with deficient snapshots. The effectiveness and robustness of the Wasserstein processor are demonstrated using acoustic simulation and the event S5 of the SWellEx-96 experiment data, exhibiting correct localization statistics and a notable reduction in ambiguity. Additionally, this paper presents an approach to derive the averaged Bartlett processor by evaluating the Wasserstein metric between two Dirac measures, providing an innovative perspective for MFP. [ABSTRACT FROM AUTHOR]

In this paper, we introduce variable Gaussian Besov-Lipschitz Bα p(p);q(γ)(d) and Triebel-Lizorkin spaces Fα p(γ);q(γ)(d); i.e., Gaussian Besov-Lipschitz and Triebel-Lizorkin spaces with variable exponents p(γ) and q(γ), under certain regularity conditions on the functions p(γ) and q(γ). The condition on p(γ) is associated with the Gaussian measure and was introduced in [3]. Trivially, they include the Gaussian Besov-Lipschitz Bα p;q(d) and Triebel-Lizorkin spaces Fα p;q(d) for p; q constants, which were introduced and studied in [10]. We consider some inclusion relations of those spaces and finally prove some interpolation results for them. [ABSTRACT FROM AUTHOR]

We study the two‐dimensional Anisotropic KPZ equation (AKPZ) formally given by ∂tH=12ΔH+λ((∂1H)2−(∂2H)2)+ξ,$$\begin{equation*} \hspace*{3.4pc}\partial _t H=\frac{1}{2}\Delta H+\lambda ((\partial _1 H)^2-(\partial _2 H)^2)+\xi , \end{equation*}$$where ξ is a space‐time white noise and λ is a strictly positive constant. While the classical two‐dimensional KPZ equation, whose nonlinearity is |∇H|2=(∂1H)2+(∂2H)2$|\nabla H|^2=(\partial _1 H)^2+(\partial _2 H)^2$, can be linearised via the Cole‐Hopf transformation, this is not the case for AKPZ. We prove that the stationary solution to AKPZ (whose invariant measure is the Gaussian Free Field (GFF)) is superdiffusive: its diffusion coefficient diverges for large times as logt$\sqrt {\mathop {\mathrm{log}}\nolimits t}$ up to loglogt$\mathop {\mathrm{log}}\nolimits \mathop {\mathrm{log}}\nolimits t$ corrections, in a Tauberian sense. Morally, this says that the correlation length grows with time like t1/2×(logt)1/4$t^{1/2}\times (\mathop {\mathrm{log}}\nolimits t)^{1/4}$. Moreover, we show that if the process is rescaled diffusively (t→t/ε2,x→x/ε,ε→0$t\rightarrow t/\varepsilon ^2, x\rightarrow x/\varepsilon , \varepsilon \rightarrow 0$), then it evolves non‐trivially already on time‐scales of order approximately 1/|logε|≪1$1/\sqrt {|\mathop {\mathrm{log}}\nolimits \varepsilon |}\ll 1$. Both claims hold as soon as the coefficient λ of the nonlinearity is non‐zero. These results are in contrast with the belief, common in the mathematics community, that the AKPZ equation is diffusive at large scales and, under simple diffusive scaling, converges to the two‐dimensional Stochastic Heat Equation (2dSHE) with additive noise (i.e., the case λ=0$\lambda =0$). [ABSTRACT FROM AUTHOR]

The quality of medical images is critical for accurate diagnosis. This paper introduces a novel Quantum-behaved Arithmetic Optimization Algorithm (QAOA) for medical images. A mutation operator with Gaussian probability distribution is used in the proposed QAOA as a powerful strategy to enhance QAOA performance in preventing premature convergence to local optima. Gaussian QAOA (GQAOA) is tailored for medical image enhancement and hybridized with Contrast Limited Adaptive Histogram Equalization (CLAHE) to boost the information contents and details of medical images. GQAOA computes the optimal clip limit and other parameters of CLAHE using a new multi-objective fitness function. A combination of five image quality measurements including contrast, information entropy, edge information, Structural Similarity Index Measure (SSIM), and sharpness is suggested as an efficient fitness function to help the proposed framework produce good results. A comparative study is conducted with well-known histogram-based process techniques and state-of-art methods to demonstrate the efficiency of the suggested algorithm. The experimental results prove that the suggested approach performs better than the most current well-established enhancement strategies in the terms of visual interpretation, information entropy, SSIM, Peak Signal to Noise Ratio (PSNR), Naturalness Image Quality Evaluator (NIQE), Absolute Mean Brightness Error (AMBE), and Quality Index (QI) metrics. [ABSTRACT FROM AUTHOR]

We study the |$k$| -nearest neighbour classifier (⁠|$k$| -NN) of probability measures under the Wasserstein distance. We show that the |$k$| -NN classifier is not universally consistent on the space of measures supported in |$(0,1)$|⁠. As any Euclidean ball contains a copy of |$(0,1)$|⁠ , one should not expect to obtain universal consistency without some restriction on the base metric space, or the Wasserstein space itself. To this end, via the notion of |$\sigma $| -finite metric dimension, we show that the |$k$| -NN classifier is universally consistent on spaces of discrete measures (and more generally, |$\sigma $| -finite uniformly discrete measures) with rational mass. In addition, by studying the geodesic structures of the Wasserstein spaces for |$p=1$| and |$p=2$|⁠ , we show that the |$k$| -NN classifier is universally consistent on spaces of measures supported on a finite set, the space of Gaussian measures and spaces of measures with finite wavelet series densities. [ABSTRACT FROM AUTHOR]

In this paper, we study the probabilistic Gel'fand  N , δ -width of multivariate Sobolev spaces  M W 2 r T d  with mixed derivative that are equipped with Gaussian measure  μ  in  L q T d . The sharp asymptotic estimates are determined by employing the discretization method. [ABSTRACT FROM AUTHOR]

We show that for any even log-concave probability measure \mu on \mathbb {R}^n, any pair of symmetric convex sets K and L, and any \lambda \in [0,1], \begin{equation*} \mu ((1-\lambda) K+\lambda L)^{c_n}\geq (1-\lambda) \mu (K)^{c_n}+\lambda \mu (L)^{c_n}, \end{equation*} where c_n\geq n^{-4-o(1)}. This constitutes progress towards the dimensional Brunn-Minkowski conjecture (see Richard J. Gardner and Artem Zvavitch [Tran. Amer. Math. Soc. 362 (2010), pp. 5333–5353]; Andrea Colesanti, Galyna V. Livshyts, Arnaud Marsiglietti [J. Funct. Anal. 273 (2017), pp. 1120–1139]). Moreover, our bound improves for various special classes of log-concave measures. [ABSTRACT FROM AUTHOR]

There are some previous works on designing efficient and high-order numerical methods of density estimation for stochastic partial differential equation (SPDE) driven by multivariate Gaussian random variables. They mostly focus on proposing numerical methods of density estimation for SPDE with independent random variables and rarely research density estimation for SPDE is driven by multivariate Gaussian random variables. In this paper, we propose a high-order algorithm of gPC-based density estimation where SPDE driven by multivariate Gaussian random variables. Our main techniques are (1) we build a new multivariate orthogonal basis by adopting the Gauss–Schmidt orthogonalization; (2) with the newly constructed orthogonal basis in hand, we first assume the unknown function in the SPDE has the stochastic general polynomial chaos (gPC) expansion, second implement the stochastic gPC expansion for the SPDE in the multivariate Gaussian measure space, and third we obtain and numerical calculation deterministic differential equations for the coefficients of the expansion; (3) we used high-order algorithm of gPC-based for density estimation and moment estimation. We apply the newly proposed numerical method to a known random function, stochastic 1D wave equation, and stochastic 2D Schnakenberg model, respectively. All the presented stochastic equations are driven by bivariate Gaussian random variables. The efficiency is compared with the Monte-Carlo method based on the known random function. [ABSTRACT FROM AUTHOR]

In the study of femtoscopic correlations in high-energy physics, besides Bose–Einstein correlations, one has to take final-state interactions into account. Amongst them, Coulomb interactions play a prominent role in the case of charged particles. Recent measurements have shown that in heavy-ion collisions, Bose–Einstein correlations can be best described by Lévy-type sources instead of the more common Gaussian assumption. Furthermore, three-dimensional measurements have indicated that, depending on the choice of frame, a deviation from spherical symmetry observed under the assumption of Gaussian source functions persists in the case of Lévy-type sources. To clarify such three-dimensional Lévy-type correlation measurements, it is thus important to study the effect of Coulomb interactions in the case of non-spherical Lévy sources. We calculated the Coulomb correction factor numerically in the case of such a source function for assorted kinematic domains and parameter values using the Metropolis–Hastings algorithm and compared our results with previous methods to treat Coulomb interactions in the presence of Lévy sources. [ABSTRACT FROM AUTHOR]