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This article is concerned with the approximation and expressive powers of deep neural networks. This is an active research area currently producing many interesting papers. The results most commonly found in the literature prove that neural networks approximate functions with classical smoothness to the same accuracy as classical linear methods of approximation, e.g., approximation by polynomials or by piecewise polynomials on prescribed partitions. However, approximation by neural networks depending on n parameters is a form of nonlinear approximation and as such should be compared with other nonlinear methods such as variable knot splines or n-term approximation from dictionaries. The performance of neural networks in targeted applications such as machine learning indicate that they actually possess even greater approximation power than these traditional methods of nonlinear approximation. The main results of this article prove that this is indeed the case. This is done by exhibiting large classes of functions which can be efficiently captured by neural networks where classical nonlinear methods fall short of the task. The present article purposefully limits itself to studying the approximation of univariate functions by ReLU networks. Many generalizations to functions of several variables and other activation functions can be envisioned. However, even in this simplest of settings considered here, a theory that completely quantifies the approximation power of neural networks is still lacking.

BackgroundSevere acute respiratory syndrome coronavirus 2 (SARS-CoV-2) infection in people with cystic fibrosis (pwCF) can lead to severe outcomes.MethodsIn this observational study, the European Cystic Fibrosis Society Patient Registry collected data on pwCF and SARS-CoV-2 infection to estimate incidence, describe clinical presentation and investigate factors associated with severe outcomes using multivariable analysis.ResultsUp to December 31, 2020, 26 countries reported information on 828 pwCF and SARS-CoV-2 infection. Incidence was 17.2 per 1000 pwCF (95% CI: 16.0–18.4). Median age was 24 years, 48.4% were male and 9.4% had lung transplants. SARS-CoV-2 incidence was higher in lung-transplanted (28.6; 95% CI: 22.7–35.5) versus non-lung-transplanted pwCF (16.6; 95% CI: 15.4–17.8) (p≤0.001).SARS-CoV-2 infection caused symptomatic illness in 75.7%. Factors associated with symptomatic SARS-CoV-2 infection were age >40 years, at least one F508del mutation and pancreatic insufficiency.Overall, 23.7% of pwCF were admitted to hospital, 2.5% of those to intensive care, and regretfully 11 (1.4%) died. Hospitalisation, oxygen therapy, intensive care, respiratory support and death were 2- to 6-fold more frequent in lung-transplanted versus non-lung-transplanted pwCF.Factors associated with hospitalisation and oxygen therapy were lung transplantation, cystic fibrosis-related diabetes (CFRD), moderate or severe lung disease and azithromycin use (often considered a surrogate marker for Pseudomonas aeruginosa infection and poorer lung function).ConclusionSARS-CoV-2 infection yielded high morbidity and hospitalisation in pwCF. PwCF with forced expiratory volume in 1 s

(1) Background: Several recent studies on the clinical value of spirometry indexes demonstrated high sensitivity of FEF(25-75) as a marker of bronchial obstruction in asthmatics with normal baseline spirometry. Our study aims to evaluate the clinical value of maximal mid-expiratory flow in children with asthma. (2) Methods: For two years, 257 children were included - 211 with asthma and 46 healthy controls. Pre- and post-bronchodilator spirometry, atopic status determination and asthma control assessment were performed. (3) Results: The small airway obstruction (SAO) group (FEV(1)≥80%, ММEF(25/75)80%, MMEF(25/75)≥65%) (Р

Background: Viral infections can cause significant morbidity in cystic fibrosis (CF). The current Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2) pandemic could therefore have a serious impact on the health of people with CF (pwCF). Methods: We used the 38-country European Cystic Fibrosis Society Patient Registry (ECFSPR) to collect case data about pwCF and SARS-CoV-2 infection. Results: Up to 30 June 2020, 16 countries reported 130 SARS-CoV-2 cases in people with CF, yielding an incidence of 2.70/10 0 0 pwCF. Incidence was higher in lung-transplanted patients (n = 23) versus non transplanted patients (n = 107) (8.43 versus 2.36 cases/10 0 0). Incidence was higher in pwCF versus the age-matched general population in the age groups < 15, 15-24, and 25-49 years (p < 0.001), with similar trends for pwCF with and without lung transplant. Compared to the general population, pwCF (regardless of transplantation status) had significantly higher rates of admission to hospital for all age groups with available data, and higher rates of intensive care, although not statistically significant. Most pwCF recovered (96.2%), however 5 died, of whom 3 were lung transplant recipients. The case fatality rate for pwCF (3.85%, 95% CI: 1.26-8.75) was non-significantly lower than that of the general population (7.46%; p = 0.133). Conclusions: SARS-CoV-2 infection can result in severe illness and death for pwCF, even for younger patients and especially for lung transplant recipients. PwCF should continue to shield from infection and should be prioritized for vaccination. (c) 2021 The Authors. Published by Elsevier B.V. on behalf of European Cystic Fibrosis Society. This is an open access article under the CC BY-NC-ND license ( http://creativecommons.org/licenses/by-nc-nd/4.0/ )

Th17 lymphocytes are now widely believed to be critical in various chronic pulmonary diseases. However, there is still a small number of investigations regarding children. We aimed to assess the percentage of Th17 lymphocytes and IL-17A in peripheral blood of children with chronic obstructive lung diseases.We included a total of 42 children: 20 with bronchial asthma (BA), 12 with cystic fibrosis (CF) and 10 healthy children without a history of allergies, aged 4-17 years. Th17 cells (CD3+CD4+CD161+CCR6+) were determined in peripheral blood by flow cytometry. The concentration of serum IL-17A was measured by ELISA.The BA patients had a significantly higher percentage of Th17 (12.40±1.16%) compared to the CF children (7.64±0.87%, p=0.0035) and healthy (7.25±0.45%, p=0.008). Stratifying the BA group, we found higher levels of Th17 in patients with severe BA (p=0.03), whereas patients with moderate BA had Th17 cells close to those in CF and healthy children. We found that patients with better control of BA had Th17 closer to those with CF (p=0.98) than BA children with poor control (p0.001) (post hoc, Bonferroni correction). CF patients with concomitant P. aeruginosa infection showed slightly higher percentages of Th17 cells than those without infection (8.08±3.09% vs. 6.25±2.42%, p=0.294).The percentage of Th17 cells was significantly increased in the peripheral blood of children with severe BA compared to the children with moderate BA, which suggests that the former could possibly benefit from future target therapies.

A numerical algorithm is presented for computing average global temperature (or other quantities of interest such as average precipitation) from measurements taken at speci_ed locations and times. The algorithm is proven to be in a certain sense optimal. The analysis of the optimal algorithm provides a sharp a priori bound on the error between the computed value and the true average global temperature. This a priori bound involves a computable compatibility constant which assesses the quality of the measurements for the chosen model. The optimal algorithm is constructed by solving a convex minimization problem and involves a set of functions selected a priori in relation to the model. It is shown that the solution promotes sparsity and hence utilizes a smaller number of well-chosen data sites than those provided. The algorithm is then applied to canonical data sets and mathematically generic models for the computation of average temperature and average precipitation over given regions and given time intervals. A comparison is provided between the proposed algorithms and existing methods.

Pulmonary rehabilitation is a key component in cystic fibrosis care. This review summarizes the recent evidence in the area of pulmonary rehabilitation for cystic fibrosis in the form of questions and answers regarding interventions, indications, benefits and risks of pulmonary rehabilitation. Pulmonary rehabilitation includes airway clearance techniques, exercise training, education and behaviour change and can improve patients’ exercise capacity, muscle strength, quality of life and nutritional status. Airway clearance techniques have beneficial effects for clearing mucous. Over the past years, evidence for the beneficial effects of exercise training on exercise capacity and overall lung health is growing. In cystic fibrosis, multiple factors result in reduced exercise capacity. All modalities of pulmonary rehabilitation should be offered to patients with cystic fibrosis, as the benefits in most cases outweigh the risks, though the optimal regimens need to be yet defined.

In the desire to quantify the success of neural networks in deep learning and other applications, there is a great interest in understanding which functions are efficiently approximated by the outputs of neural networks. By now, there exists a variety of results which show that a wide range of functions can be approximated with sometimes surprising accuracy by these outputs. For example, it is known that the set of functions that can be approximated with exponential accuracy (in terms of the number of parameters used) includes, on one hand, very smooth functions such as polynomials and analytic functions (see e.g. \cite{E,S,Y}) and, on the other hand, very rough functions such as the Weierstrass function (see e.g. \cite{EPGB,DDFHP}), which is nowhere differentiable. In this paper, we add to the latter class of rough functions by showing that it also includes refinable functions. Namely, we show that refinable functions are approximated by the outputs of deep ReLU networks with a fixed width and increasing depth with accuracy exponential in terms of their number of parameters. Our results apply to functions used in the standard construction of wavelets as well as to functions constructed via subdivision algorithms in Computer Aided Geometric Design.

While it is well known that nonlinear methods of approximation can often perform dramatically better than linear methods, there are still questions on how to measure the optimal performance possible for such methods. This paper studies nonlinear methods of approximation that are compatible with numerical implementation in that they are required to be numerically stable. A measure of optimal performance, called {\em stable manifold widths}, for approximating a model class $K$ in a Banach space $X$ by stable manifold methods is introduced. Fundamental inequalities between these stable manifold widths and the entropy of $K$ are established. The effects of requiring stability in the settings of deep learning and compressed sensing are discussed.

Autosomal dominant hyper-IgE syndrome (AD-HIES) is a rare disease described in 1966. It is characterized by severe dermatitis, a peculiar face, frequent infections, extremely high levels of serum IgE and eosinophilia, all resulting from a defect in the STAT3 gene. A variety of mutations in the SH2 and DNA-binding domain have been described, and several studies have searched for associations between the severity of the clinical symptoms, laboratory findings, and the type of genetic alteration. We present two children with AD-HIES–a girl with the most common STAT3 mutation (R382W) and a boy with a rare variant (G617E) in the same gene, previously reported in only one other patient. Herein, we discuss the clinical and immunological findings in our patients, focusing on their importance on disease course and management.

Mycoplasma pneumonia(MP) is a common respiratory pathogen that produces diseases of varied severity ranging from mild upper respiratory tract infection to severe atypical pneumonia. The aim of the study was to describe the clinical features, in children admitted for respiratory problems in the clinic for the last year that were tested positive for MP infection. Methods: All children were tested for MP with serology (IgM and IgG) and PCR. Medical charts, as well as radiological findings, were reviewed for all. Results For a period of 1 year 54 children (mean age 12.3 years ± 2.23, 26 boys and 28 girls) were tested positive for MP by PCR or serology. Almost ¾ of them were admitted for pneumonia, 17% for bronchiolitis and 9% for asthma attack. Infiltrative x-ray changes were found in 46%, while 32% had interstitial changes and the rest showed marked obstruction. All patient had cough and fever (38.5◦C±2.4◦C) from 1 to 20 days and 52% had received antibiotics prior hospitalization. All patient had elevated markers for inflammation CRP (mean 29.7±16.5 mg/l) and ESR (mean 70.3 ± 17.5 mm). From the spirometry results if the children 14 of those children had FEV1 ≤ 80% predicted, while those with normal values of FEV1 had reduced values for MMEF25-75. All children were discharged asymptomatic after in hospital treatments for about 5 days. Conclusion: M. pneumoniae symptoms were more similar to infections with other respiratory viruses then the ones with typical bacteria (i.e. S.aureus or S.pneumoniae), but in over 70% there were X-ray changes suggestive for pneumonia. Proper and timely diagnosis of MP is curtail for limiting the pointless use of betalactams, which could be saved for other respiratory pathogens.

Chronic lung infection is one of the hallmarks in cystic fibrosis (CF). Respiratory cultures from CF patients usually indicate colonization most commonly with Staphylococcus aureus, followed by Pseudomonas aeruginosa and Burkholderia cepacia, all of them listed as a classic pathogen for CF. We present a case of 15-year-old patient with CF, diagnosed at the age of 2 years due to failure to thrive and recurrent lung infections. He survived severe peritonitis at the age of 5 years, when he was colonized with P.aeruginosa for the first time. Following strict intravenous (piperacillin/tazobactam or ceftazidime) and inhaled antibiotic regimens (tobramycin, colistin and azytromycin), he is P.aeruginosa free for the last 6 years. However, in his sputum samples grew constanly Achrombacter xylosoxidans. Rigourous inhaled colistin and sulfomethaxasole/trimethoprim therapy was followed until the germ was resistant to every antibiotic tested but meropenem. Last year he was on inhaled meropenem with favorable effect – no A.xylosoxidans in his sputum from more than 6 months. His lung health was relatively stable at FVE1 of about 45-50% for almost 2 years, but this summer he had experienced new exacerbation and FEV1 drop to 26%. At that moment we found Myroides odoratus in his sputum sampled. The therapy was modified. Three months later in the sputum of the patient we found only S.aureus and some fungal colonies, his FEV1 is back at 50% and he gained 2 kilos. Myroides remains an uncommon pathogen that is ubiquitous in the environment. Clinicians should remain alert to the possibility of this one or other new emerging pathogens in CF when there is a lack of response from routine treatment

Tuberculosis is a mycobacterial infection that can affect the lungs as well as other organs. The involvement of the spine, although rare, can have major consequences if not diagnosed and treated in a timely and effective manner, such as residual deformities and neurological deficits. On occasion, the atypical presentation of tuberculous spondylitis may cause a delay in treatment and therefore lead to less favorable outcomes. In this article, we present a rare case of progressed tuberculous infection involving the respiratory and musculoskeletal system in a 36-year-old patient whose main complaints were non-specific and mild, and started only two weeks before his diagnosis, despite the advanced disease.

Neural networks (NNs) are the method of choice for building learning algorithms. They are now being investigated for other numerical tasks such as solving high-dimensional partial differential equations. Their popularity stems from their empirical success on several challenging learning problems (computer chess/Go, autonomous navigation, face recognition). However, most scholars agree that a convincing theoretical explanation for this success is still lacking. Since these applications revolve around approximating an unknown function from data observations, part of the answer must involve the ability of NNs to produce accurate approximations.This article surveys the known approximation properties of the outputs of NNs with the aim of uncovering the properties that are not present in the more traditional methods of approximation used in numerical analysis, such as approximations using polynomials, wavelets, rational functions and splines. Comparisons are made with traditional approximation methods from the viewpoint of rate distortion, i.e. error versus the number of parameters used to create the approximant. Another major component in the analysis of numerical approximation is the computational time needed to construct the approximation, and this in turn is intimately connected with the stability of the approximation algorithm. So the stability of numerical approximation using NNs is a large part of the analysis put forward.The survey, for the most part, is concerned with NNs using the popular ReLU activation function. In this case the outputs of the NNs are piecewise linear functions on rather complicated partitions of the domain of f into cells that are convex polytopes. When the architecture of the NN is fixed and the parameters are allowed to vary, the set of output functions of the NN is a parametrized nonlinear manifold. It is shown that this manifold has certain space-filling properties leading to an increased ability to approximate (better rate distortion) but at the expense of numerical stability. The space filling creates the challenge to the numerical method of finding best or good parameter choices when trying to approximate.

Autosomal dominant hyper-IgE syndrome (AD-HIES) is a rare disease described in 1966. It is characterized by severe dermatitis, a peculiar face, frequent infections, extremely high levels of serum IgE and eosinophilia, all resulting from a defect in the STAT3 gene. A variety of mutations in the SH2 and DNA-binding domain have been described, and several studies have searched for associations between the severity of the clinical symptoms, laboratory findings, and the type of genetic alteration. We present two children with AD-HIES–a girl with the most common STAT3 mutation (R382W) and a boy with a rare variant (G617E) in the same gene, previously reported in only one other patient. Herein, we discuss the clinical and immunological findings in our patients, focusing on their importance on disease course and management.

The usual approach to model reduction for parametric partial differential equations (PDEs) is to construct a linear space $V_n$ which approximates well the solution manifold $\mathcal{M}$ consisting of all solutions $u(y)$ with $y$ the vector of parameters. This linear reduced model $V_n$ is then used for various tasks such as building an online forward solver for the PDE or estimating parameters from data observations. It is well understood in other problems of numerical computation that nonlinear methods such as adaptive approximation, $n$-term approximation, and certain tree-based methods may provide improved numerical efficiency. For model reduction, a nonlinear method would replace the linear space $V_n$ by a nonlinear space $\Sigma_n$. This idea has already been suggested in recent papers on model reduction where the parameter domain is decomposed into a finite number of cells and a linear space of low dimension is assigned to each cell. Up to this point, little is known in terms of performance guarantees for such a nonlinear strategy. Moreover, most numerical experiments for nonlinear model reduction use a parameter dimension of only one or two. In this work, a step is made towards a more cohesive theory for nonlinear model reduction. Framing these methods in the general setting of library approximation allows us to give a first comparison of their performance with those of standard linear approximation for any general compact set. We then turn to the study these methods for solution manifolds of parametrized elliptic PDEs. We study a very specific example of library approximation where the parameter domain is split into a finite number $N$ of rectangular cells and where different reduced affine spaces of dimension $m$ are assigned to each cell. The performance of this nonlinear procedure is analyzed from the viewpoint of accuracy of approximation versus $m$ and $N$.

Scientific problems often feature observational data received in the form $$w_1=l_1(f),\ldots $$ , $$w_m=l_m(f)$$ of known linear functionals applied to an unknown function f from some Banach space $$\mathcal {X}$$ , and it is required to either approximate f (the full approximation problem) or to estimate a quantity of interest Q(f). In typical examples, the quantities of interest can be the maximum/minimum of f or some averaged quantity such as the integral of f, while the observational data consists of point evaluations. To obtain meaningful results about such problems, it is necessary to possess additional information about f, usually as an assumption that f belongs to a certain model class $$\mathcal {K}$$ contained in $$\mathcal {X}$$ . This is precisely the framework of optimal recovery, which produced substantial investigations when the model class is a ball in a smoothness space, e.g., when it is a unit ball in Lipschitz, Sobolev, or Besov spaces. This paper is concerned with other model classes described by approximation processes, as studied in DeVore et al. [Data assimilation in Banach spaces, (To Appear)]. Its main contributions are: (1) designing implementable optimal or near-optimal algorithms for the estimation of quantities of interest, (2) constructing linear optimal or near-optimal algorithms for the full approximation of an unknown function using its point evaluations. While the existence of linear optimal algorithms for the approximation of linear functionals Q(f) is a classical result established by Smolyak, a numerically friendly procedure that performs this approximation is not generally available. In this paper, we show that in classical recovery settings, such linear optimal algorithms can be produced by constrained minimization methods. We illustrate these techniques on several examples involving the computation of integrals using point evaluation data. In addition, we show that linearization of optimal algorithms can be achieved for the full approximation problem in the important situation where the $$l_j$$ are point evaluations and $$\mathcal {X}$$ is a space of continuous functions equipped with the uniform norm. It is also revealed how quasi-interpolation theory enables the construction of linear near-optimal algorithms for the recovery of the underlying function.

We consider the problem of optimal recovery of an element $u$ of a Hilbert space $\mathcal{H}$ from $m$ measurements obtained through known linear functionals on $\mathcal{H}$. Problems of this type are well studied \cite{MRW} under an assumption that $u$ belongs to a prescribed model class, e.g. a known compact subset of $\mathcal{H}$. Motivated by reduced modeling for parametric partial differential equations, this paper considers another setting where the additional information about $u$ is in the form of how well $u$ can be approximated by a certain known subspace $V_n$ of $\mathcal{H}$ of dimension $n$, or more generally, how well $u$ can be approximated by each $k$-dimensional subspace $V_k$ of a sequence of nested subspaces $V_0\subset V_1\cdots\subset V_n$. A recovery algorithm for the one-space formulation, proposed in \cite{MPPY}, is proven here to be optimal and to have a simple formulation, if certain favorable bases are chosen to represent $V_n$ and the measurements. The major contribution of the present paper is to analyze the multi-space case for which it is shown that the set of all $u$ satisfying the given information can be described as the intersection of a family of known ellipsoids in $\mathcal{H}$. It follows that a near optimal recovery algorithm in the multi-space problem is to identify any point in this intersection which can provide a much better accuracy than in the one-space problem. Two iterative algorithms based on alternating projections are proposed for recovery in the multi-space problem. A detailed analysis of one of them provides a posteriori performance estimates for the iterates, stopping criteria, and convergence rates. Since the limit of the algorithm is a point in the intersection of the aforementioned ellipsoids, it provides a near optimal recovery for $u$., 27 pages

Background The spectrum and frequencies of CFTR mutations causing Cystic fibrosis (CF) varies among different populations in Europe, and beyond. Methods We identified 98.9% of all CFTR mutations in a representative cohort of 140 CF patients comprising 107 Bulgarian‐ (BG), 17 BG Turk‐, and 16 BG Roma cases. The compiled clinical and genotype dataset includes 110 previously analyzed patients with 30 cases currently analyzed for rare CFTR variants by massively parallel sequencing of the entire CFTR coding region and adjacent introns combined with the analysis of intra‐CFTR rearrangements. Results Altogether 53 different mutations, of which 15 newly identified in the BG CF population, were observed. Comparison of clinical and laboratory data between individual BG ethnic groups proved that BG Roma have a more severe nutritional status and are younger than other CF patients, as well as that the spectrum mutations differs between them. Conclusion This collaborative study improves genetic counselling in BG, facilitates introduction of multitier CF neonatal screening and fosters public health measures for improvement of care in the Roma CF population.

Motivated by numerical methods for solving parametric partial differential equations, this paper studies the approximation of multivariate analytic functions by algebraic polynomials. We introduce various anisotropic model classes based on Taylor expansions, and study their approximation by finite dimensional polynomial spaces $${{\mathcal {P}}}_\Lambda $$ described by lower sets $$\Lambda $$ . Given a budget n for the dimension of $${{\mathcal {P}}}_\Lambda $$ , we prove that certain lower sets $$\Lambda _n$$ , with cardinality n, provide a certifiable approximation error that is in a certain sense optimal, and that these lower sets have a simple definition in terms of simplices. Our main goal is to obtain approximation results when the number of variables d is large and even infinite, and so we concentrate almost exclusively on the case $$d=\infty $$ . We also emphasize obtaining results which hold for the full range $$n\ge 1$$ , rather than asymptotic results that only hold for n sufficiently large. In applications, one typically wants n small to comply with computational budgets.

Scoliosis is the most common abnormality of the spine with direct effects on the thoracic cage, leading to potentially severe and irreversible changes in lung function. Scoliosis has generally been associated with the development of restrictive lung disease. Due to the lateral curvature of the spine in children with scoliosis, the measured height does not correspond to the actual one and could not be used as the authoritative index for the calculation of the predicted values for both static and dynamic lung volumes.

We show that a very simple modification of the Pure Greedy Algorithm for approximating functions by sparse sums from a dictionary in a Hilbert or more generally a Banach space has optimal convergence rates on the class of convex combinations of dictionary elements

Summary Histiocytoses comprise a group of diverse diseases of unknown etiology with various clinical presentation and evolution. The underlying pathology is characterised by accumulation and infiltration of variable numbers of cells of the monocyte-macrophage line in the affected tissues and organs. Histiocytoses are divided into three major classes: Langerhans cell histiocytosis (LCH), non- Langerhans cell histiocytosis, and malignant histiocytic disorders. The term LCH (also known in the past as histiocytosis X) encompasses the following rare diseases: Eosinophilic Granuloma, Hand-Schuller-Christian disease, Letterer-Siwe disease, Hashimoto-Pritzker disease, in which accumulation of pathologic Langerhans cells (LCs) leads to tissue damage. LCs usually reside in the skin and ensure protection against infections by destroying foreign substances. LC accumulation is caused by antigen stimulation and inadequate immune response. Thus, clinical LCH manifestations range from isolated disease with mono- or multifocal bone lesions to disseminated multisystem disease. LCH is a rare disease, affecting mainly children and young smokers, aged 20-50 years. Lung involvement in LCH usually presents as a mono-system disease and is characterized by Langerhans cell granulomas (LCG) infiltrating and impairing the distal bronchioles. The definite diagnosis is based on lung biopsy of CAT selected LCG areas. So far, there is no an effective treatment, but the better understanding of the mechanisms involved in the pathogenesis of the disease would help in the development of effective therapeutic strategies in the future.

We compute semi-explicitly or recursively the weights of the generalized Gauss–Radau and Gauss–Lobatto quadratures with Jacobi weight functions.

We construct cubatures that approximate the integral of a function u over the unit ball by the linear combination of surface integrals over the unit sphere of normal derivatives of u and surface integrals of u and Δ 2 u over m spheres, centered at the origin. We derive explicitly the weights and the nodes of these cubatures, and show that they are exact for all ( 2 m + 2 ) -harmonic functions.