RECONSTRUCTION PROBLEMS OF CONVEX BODIES FROM EVEN L_(p)SURFACE AREA MEASURES

We build a computer program to reconstruct convex bodies using even L_(p)surface area measures for p≥1.Firstly,we transform the minimization problem Pi,which is equivalent to solving the even L_(p)Minkowski problem,into a convex optimization problem P4 with a finite number of constraints.This transformation makes it suitable for computational resolution.Then,we prove that the approximate solutions obtained by solving the problem P4 converge to the theoretical solution when N and k are sufficiently large.Finally,based on the convex optimization problem P_(4),we provide an algorithm for reconstructing convex bodies from even L_(p)surface area measures,and present several examples implemented using MATLAB.

Algorithmic approaches to clonal reconstruction in heterogeneous cell populations

Background:The reconstruction of clonal haplotypes and their evolutionary history in evolving populations is a common problem in both microbial evolutionary biology and cancer biology.The clonal theory of evolution provides a theoretical framework for modeling the evolution of clones.Results:In this paper,we review the theoretical framework and assumptions over which the clonal reconstruction problem is formulated.We formally define the problem and then discuss the complexity and solution space of the problem.Various methods have been proposed to find the phylogeny that best explains the observed data.We categorize these methods based on the type of input data that they use(space-resolved or time-resolved),and also based on their computational formulation as either combinatorial or probabilistic.It is crucial to understand the different types of input data because each provides essential but distinct information for drastically reducing the solution space of the clonal reconstruction problem.Complementary information provided by single cell sequencing or from whole genome sequencing of randomly isolated clones can also improve the accuracy of clonal reconstruction.We briefly review the existing algorithms and their relationships.Finally we summarize the tools that are developed for either directly solving the clonal reconstruction problem or a related computational problem.Conclusions:In this review,we discuss the various formulations of the problem of inferring the clonal evolutionary history from allele frequeny data,review existing algorithms and catergorize them according to their problem formulation and solution approaches.We note that most of the available clonal inference algorithms were developed for elucidating tumor evolution whereas clonal reconstruction for unicellular genomes are less addressed.We conclude the review by discussing more open problems such as the lack of benchmark datasets and comparison of performance between available tools.

Utilizing BP neural networks to accurately reconstruct the tritium depth profile in materials for BIXS

β-ray-induced X-ray spectroscopy(BIXS)is a promising method for tritium detection in solid materials because of its unique advantages,such as large detection depth,nondestructive testing capabilities,and low requirements for sample preparation.However,high-accuracy reconstruction of the tritium depth profile remains a significant challenge for this technique.In this study,a novel reconstruction method based on a backpropagation(BP)neural network algorithm that demonstrates high accuracy,broad applicability,and robust noise resistance is proposed.The average reconstruction error calculated using the BP network(8.0%)was much lower than that obtained using traditional numerical methods(26.5%).In addition,the BP method can accurately reconstruct BIX spectra of samples with an unknown range of tritium and exhibits wide applicability to spectra with various tritium distributions.Furthermore,the BP network demonstrates superior accuracy and stability compared to numerical methods when reconstructing the spectra,with a relative uncertainty ranging from 0 to 10%.This study highlights the advantages of BP networks in accurately reconstructing the tritium depth profile from BIXS and promotes their further application in tritium detection.