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Many combinatorial optimization problems can be formulated as the search for a subgraph that satisfies certain properties and minimizes the total weight. We assume here that the vertices correspond to points in a metric space and can take any position in given uncertainty sets. Then, the cost function to be minimized is the sum of the distances for the worst positions of the vertices in their uncertainty sets. We propose two types of polynomial-time approximation algorithms. The first one relies on solving a deterministic counterpart of the problem where the uncertain distances are replaced with maximum pairwise distances. We study in details the resulting approximation ratio, which depends on the structure of the feasible subgraphs and whether the metric space is Ptolemaic or not. The second algorithm is a fully-polynomial time approximation scheme for the special case of $s-t$ paths.

Differential abundance analysis is a key component of microbiome studies. While dozens of methods for it exist, currently, there is no consensus on the preferred methods. Correctness of results in differential abundance analysis is an ambiguous concept that cannot be evaluated without employing simulated data, but we argue that consistency of results across datasets should be considered as an essential quality of a well-performing method. We compared the performance of 14 differential abundance analysis methods employing datasets from 54 taxonomic profiling studies based on 16S rRNA gene or shotgun sequencing. For each method, we examined how the results replicated between random partitions of each dataset and between datasets from independent studies. While certain methods showed good consistency, some widely used methods were observed to produce a substantial number of conflicting findings. Overall, the highest consistency without unnecessary reduction in sensitivity was attained by analyzing relative abundances with a non-parametric method (Wilcoxon test or ordinal regression model) or linear regression (MaAsLin2). Comparable performance was also attained by analyzing presence/absence of taxa with logistic regression.

We present LinApart, a routine designed for efficiently performing the univariate partial fraction decomposition of large symbolic expressions. Our method is based on an explicit closed formula for the decomposition of rational functions with fully factorized denominators. We provide implementations in both the Wolfram Mathematica and C languages, made available at //github.com/fekeshazy/LinApart . The routine can provide very significant performance gains over available tools such as the Apart command in Mathematica.

This paper studies the convergence of a spatial semidiscretization of a three-dimensional stochastic Allen-Cahn equation with multiplicative noise. For non-smooth initial values, the regularity of the mild solution is investigated, and an error estimate is derived with the spatial $ L^2 $-norm. For smooth initial values, two error estimates with the general spatial $ L^q $-norms are established.

Multi-target linear shrinkage is an extension of the standard single-target linear shrinkage for covariance estimation. We combine several constant matrices - the targets - with the sample covariance matrix. We derive the oracle and a \textit{bona fide} multi-target linear shrinkage estimator with exact and empirical mean. In both settings, we proved its convergence towards the oracle under Kolmogorov asymptotics. Finally, we show empirically that it outperforms other standard estimators in various situations.

The problems of optimal recovering univariate functions and their derivatives are studied. To solve these problems, two variants of the truncation method are constructed, which are order-optimal both in the sense of accuracy and in terms of the amount of involved Galerkin information. For numerical summation, it has been established how the parameters characterizing the problem being solved affect its stability.

Simulation of the monodomain equation, crucial for modeling the heart's electrical activity, faces scalability limits when traditional numerical methods only parallelize in space. To optimize the use of large multi-processor computers by distributing the computational load more effectively, time parallelization is essential. We introduce a high-order parallel-in-time method addressing the substantial computational challenges posed by the stiff, multiscale, and nonlinear nature of cardiac dynamics. Our method combines the semi-implicit and exponential spectral deferred correction methods, yielding a hybrid method that is extended to parallel-in-time employing the PFASST framework. We thoroughly evaluate the stability, accuracy, and robustness of the proposed parallel-in-time method through extensive numerical experiments, using practical ionic models such as the ten-Tusscher-Panfilov. The results underscore the method's potential to significantly enhance real-time and high-fidelity simulations in biomedical research and clinical applications.

Predictive posterior densities (PPDs) are of interest in approximate Bayesian inference. Typically, these are estimated by simple Monte Carlo (MC) averages using samples from the approximate posterior. We observe that the signal-to-noise ratio (SNR) of such estimators can be extremely low. An analysis for exact inference reveals SNR decays exponentially as there is an increase in (a) the mismatch between training and test data, (b) the dimensionality of the latent space, or (c) the size of the test data relative to the training data. Further analysis extends these results to approximate inference. To remedy the low SNR problem, we propose replacing simple MC sampling with importance sampling using a proposal distribution optimized at test time on a variational proxy for the SNR and demonstrate that this yields greatly improved estimates.

Conventional computing paradigm struggles to fulfill the rapidly growing demands from emerging applications, especially those for machine intelligence, because much of the power and energy is consumed by constant data transfers between logic and memory modules. A new paradigm, called "computational random-access memory (CRAM)" has emerged to address this fundamental limitation. CRAM performs logic operations directly using the memory cells themselves, without having the data ever leave the memory. The energy and performance benefits of CRAM for both conventional and emerging applications have been well established by prior numerical studies. However, there lacks an experimental demonstration and study of CRAM to evaluate its computation accuracy, which is a realistic and application-critical metrics for its technological feasibility and competitiveness. In this work, a CRAM array based on magnetic tunnel junctions (MTJs) is experimentally demonstrated. First, basic memory operations as well as 2-, 3-, and 5-input logic operations are studied. Then, a 1-bit full adder with two different designs is demonstrated. Based on the experimental results, a suite of modeling has been developed to characterize the accuracy of CRAM computation. Scalar addition, multiplication, and matrix multiplication, which are essential building blocks for many conventional and machine intelligence applications, are evaluated and show promising accuracy performance. With the confirmation of MTJ-based CRAM's accuracy, there is a strong case that this technology will have a significant impact on power- and energy-demanding applications of machine intelligence.