Optimal recovery of operator sequences

In this paper we solve two problems of optimal recovery based on information given with an error. First is the problem of optimal recovery of the class $W^T_q = \{(t_1h_1,t_2h_2,\ldots)\,\colon \,\|h\|_{\ell_q}\le 1\}$, where $1\le q < \infty$ and $t_1\ge t_2\ge \ldots \ge 0$ are given, in the space $\ell_q$. Information available about a sequence $x\in W^T_q$ is provided either (i) by an element $y\in\mathbb{R}^n$, $n\in\mathbb{N}$, whose distance to the first $n$ coordinates $\left(x_1,\ldots,x_n\right)$ of $x$ in the space $\ell_r^n$, $0 < r \le \infty$, does not exceed given $\varepsilon\ge 0$, or (ii) by a sequence $y\in\ell_\infty$ whose distance to $x$ in the space $\ell_r$ does not exceed $\varepsilon$. We show that the optimal method of recovery in this problem is either operator $\Phi^*_m$ with some $m\in\mathbb{Z}_+$ ($m\le n$ in case $y\in\ell^n_r$), where \smallskip\centerline{$\displaystyle \Phi^*_m(y) = \Big\{y_1\left(1 - \frac{t_{m+1}^q}{t_{1}^q}\Big),\ldots,y_m\Big(1 - \frac{t_{m+1}^q}{t_{m}^q}\Big),0,\ldots\right\},\quad y\in\mathbb{R}^n\text{ or } y\in\ell_\infty,$} \smallskip\noior convex combination $(1-\lambda) \Phi^*_{m+1} + \lambda\Phi^*_{m}$. The second one is the problem of optimal recovery of the scalar product operator acting on the Cartesian product $W^{T,S}_{p,q}$ of classes $W^T_p$ and $W^S_q$, where $1 < p,q < \infty$, $\frac{1}{p} + \frac{1}{q} = 1$ and $s_1\ge s_2\ge \ldots \ge 0$ are given. Information available about elements $x\in W^T_p$ and $y\in W^S_q$ is provided by elements $z,w\in \mathbb{R}^n$ such that the distance between vectors $\left(x_1y_1, x_2y_2,\ldots,x_ny_n\right)$ and $\left(z_1w_1,\ldots,z_nw_n\right)$ in the space $\ell_r^n$ does not exceed $\varepsilon$. We show that the optimal method of recovery is delivered either by operator $\Psi^*_m$ with some $m\in\{0,1,\ldots,n\}$, where \smallskip\centerline{$\displaystyle \Psi^*_m = \sum_{k=1}^m z_kw_k\Big(1 - \frac{t_{m+1}s_{m+1}}{t_ks_k}\Big),\quad z,w\in\mathbb{R}^n,$} \smallskip\noior by convex combination $(1-\lambda)\Psi^*_{m+1} + \lambda\Psi^*_{m}$. As an application of our results we consider the problem of optimal recovery of classes in Hilbert spaces by the Fourier coefficients of its elements known with an error measured in the space $\ell_p$ with $p > 2$.

Management of Acute Traumatic Spinal Cord Injury: A Review of the Literature

Traumatic spinal cord injury (TSCI) is a debilitating disease that poses significant functional and economic burden on both the individual and societal levels. Prognosis is dependent on the extent of the spinal injury and the severity of neurological dysfunction. If not treated rapidly, patients with TSCI can suffer further secondary damage and experience escalating disability and complications. It is important to quickly assess the patient to identify the location and severity of injury to make a decision to pursue a surgical and/or conservative management. However, there are many conditions that factor into the management of TSCI patients, ranging from the initial presentation of the patient to long-term care for optimal recovery. Here, we provide a comprehensive review of the etiologies of spinal cord injury and the complications that may arise, and present an algorithm to aid in the management of TSCI.

Recovery of continuous functions of two variables from their Fourier coefficients known with error

In this paper, we continue to study the classical problem of optimal recovery for the classes of continuous functions. The investigated classes $W^{\psi}_{2,p}$, $1 \leq p < \infty$, consist of functions that are given in terms of generalized smoothness $\psi$. Namely, we consider the two-dimensional case which complements the recent results from [Res. Math. 2020, 28 (2), 24-34] for the classes $W^{\psi}_p$ of univariate functions. As to available information, we are given the noisy Fourier coefficients $y^{\delta}_{i,j} = y_{i,j} + \delta \xi_{i,j}$, $\delta \in (0,1)$, $i,j = 1,2, \dots$, of functions with respect to certain orthonormal system $\{ \varphi_{i,j} \}_{i,j=1}^{\infty}$, where the noise level is small in the sense of the norm of the space $l_p$, $1 \leq p < \infty$, of double sequences $\xi=( \xi_{i,j} )_{i,j=1}^{\infty}$ of real numbers. As a recovery method, we use the so-called $\Lambda$-method of summation given by certain two-dimensional triangular numerical matrix $\Lambda = \{ \lambda_{i,j}^n \}_{i,j=1}^n$, where $n$ is a natural number associated with the sequence $\psi$ that define smoothness of the investigated functions. The recovery error is estimated in the norm of the space $C ([0,1]^2)$ of continuous on $[0,1]^2$ functions. We showed, that for $1\leq p < \infty$, under the respective assumptions on the smoothness parameter $\psi$ and the elements of the matrix $\Lambda$, it holds \[ \Delta( W^{\psi}_{2,p}, \Lambda, l_p)= \sup\limits_{ y \in W^{\psi}_{2,p} } \sup\limits_{\| \xi \|_{l_p} \leq 1} \Big\| y - \sum\limits_{i=1}^{n} \sum\limits_{j=1}^{n} \lambda_{i,j}^n ( y_{i,j} + \delta \xi_{i,j}) \varphi_{i,j} \Big\|_{C ([0,1]^2)} \ll \frac{ n^{\beta + 1 - 1/{p}}}{\psi(n)}.\]

Long-term outcomes following severe COVID-19 infection: a propensity matched cohort study

BackgroundThere are limited data describing the long-term outcomes of severe COVID-19. We aimed to evaluate the long-term psychosocial and physical consequences of severe COVID-19 for patients.MethodsWe conducted a multicentre observational cohort study; between 3 and 7 months posthospital discharge, patients who had been admitted to critical care due to severe COVID-19 were invited to an established recovery service. Standardised questionnaires concerning emotional, physical and social recovery, including information on employment, were completed by patients. Using propensity score matching, we explored outcomes between patients admitted to critical care with and without COVID-19, using data from the same recovery programme.ResultsBetween July 2020 and December 2020, 93 patients who had been admitted to critical with COVID-19 participated. Emotional dysfunction was common: 46.2% of patients had symptoms of anxiety and 34.4% symptoms of depression. At follow-up 53.7% of previously employed patients had returned to employment; there was a significant difference in return to employment across the socio-economic gradient, with lower numbers of patients from the most deprived areas returning to employment (p=0.03). 91 (97.8%) COVID-19 patients were matched with 91 non-COVID-19 patients. There were no significant differences in any measured outcomes between the two cohorts.InterpretationEmotional and social problems are common in survivors of severe COVID-19 infection. Coordinated rehabilitation is required to ensure patients make an optimal recovery.

The transfer temperature from slow cooling to cryogenic storage is critical for optimal recovery of cryopreserved mammalian cells

Cryopreservation is a key step for the effective delivery of many cell therapies and for the maintenance of biological materials for research. The preservation process must be carefully controlled to ensure maximum, post-thaw recovery using cooling rates slow enough to allow time for cells to cryodehydrate sufficiently to avoid lethal intracellular ice. This study focuses on determining the temperature necessary at the end of controlled slow cooling before transfer to cryogenic storage which ensures optimal recovery of the processed cell samples. Using nucleated, mammalian cell lines derived from liver (HepG2), ovary (CHO) and bone tissue (MG63) this study has shown that cooling must be controlled to -40°C before transfer to long term storage to ensure optimal cell recovery. No further advantage was seen by controlling cooling to lower temperatures. These results are consistent with collected differential scanning calorimetry data, that indicated the cells underwent an intracellular, colloidal glass transition between -49 and -59°C (Tg’i) in the presence of the cryoprotective agent dimethyl sulfoxide (DMSO). The glass forms at the point of maximum cryodehydration and no further cellular dehydration is possible. At this point the risk of lethal intracellular ice forming on transfer to ultra-low temperature storage is eliminated. In practice it may not be necessary to continue slow cooling to below this temperature as optimal recovery at -40°C indicates that the cells have become sufficiently dehydrated to avoid further, significant damage when transferred into ultra-low temperature storage.

Easy and effective analytical method of carbendazim, dimethomorph, and fenoxanil from Protaetia brevitarsis seulensis using LC-MS/MS

Traditionally in Korea, Protaetia brevitarsis seulensis (white-spotted flower chafer) has been used as a medicine, and recently has attracted increased attention due to its antithrombotic efficacy. Some of spent mushroom compost or fermented oak sawdust, a feedstock for P. brevitarsis, were contaminated with three fungicides, carbendazim, dimethomorph, and fenoxanil, which could be transferred to the insect. This study was aimed to optimize a simple extraction method combined with liquid chromatography tandem mass spectrometry and apply it to the real samples. After the pulverized samples (5 g) were extracted with acetonitrile (10 mL) and formic acid (100 μL), fat and lipids in the samples were slowly precipitated at -20°C for 24 hours. After eight different clean-up methods were investigated, the mixture of 150 mg MgSO4/25 mg PSA/25 mg C18 was selected due to optimal recovery of the target compounds. Recovery (77.9%‒80.8% for carbendazim, 111.2%‒116.7% for dimethomorph, and 111.9%‒112.5% for fenoxanil) was achieved with reasonable relative standard deviation (<5.5%) The analytical method developed in this study was used to analyze three compounds in the 24 insect samples donated by the insect farm owners but no target compounds were detected. These results can provide important data for establishing the pesticide safety standards for P. brevitarsis before the medical applications.

Extracellular Vesicles in Regeneration and Rehabilitation Recovery after Stroke

Patients that survive after a stroke event may present disabilities that can persist for a long time or permanently after it. If stroke prevention fails, the prompt and combinatorial intervention with pharmacological and rehabilitation therapy is pivotal for the optimal recovery of patients and the reduction of disabilities. In the present review, we summarize some key features of the complex events that occur in the brain during and after the stroke event, with a special focus on extracellular vesicles (EVs) and their role as both carriers of biomarkers and potential therapeutics. EVs have already demonstrated their ability to be used for diagnostic purposes for multiple brain disorders and could represent valuable tools to track the regenerative and inflammatory processes occurring in the injured brain after stroke. Last, but not least, the use of artificial or stem cell-derived EVs were proved to be effective in stimulating brain remodeling and ameliorating recovery after stroke. Still, effective biomarkers of recovery are needed to design robust trials for the validation of innovative therapeutic strategies, such as regenerative rehabilitation approaches.

Tailored Polymer-Based Selective Extraction of Lipid Mediators from Biological Samples

Lipid mediators, small molecules involved in regulating inflammation and its resolution, are a class of lipids of wide interest as their levels in blood and tissues may be used to monitor health and disease states or the effect of new treatments. These molecules are present at low levels in biological samples, and an enrichment step is often needed for their detection. We describe a rapid and selective method that uses new low-cost molecularly imprinted (MIP) and non-imprinted (NIP) polymeric sorbents for the extraction of lipid mediators from plasma and tissue samples. The extraction process was carried out in solid-phase extraction (SPE) cartridges, manually packed with the sorbents. After extraction, lipid mediators were quantified by liquid chromatography–tandem mass spectrometry (LC–MSMS). Various parameters affecting the extraction efficiency were evaluated to achieve optimal recovery and to reduce non-specific interactions. Preliminary tests showed that MIPs, designed using the prostaglandin biosynthetic precursor arachidonic acid, could effectively enrich prostaglandins and structurally related molecules. However, for other lipid mediators, MIP and NIP displayed comparable recoveries. Under optimized conditions, the recoveries of synthetic standards ranged from 62% to 100%. This new extraction method was applied to the determination of the lipid mediators concentration in human plasma and mouse tissues and compared to other methods based on commercially available cartridges. In general, the methods showed comparable performances. In terms of structural specificity, our newly synthesized materials accomplished better retention of prostaglandins (PGs), hydroxydocosahexaenoic acid (HDoHE), HEPE, hydroxyeicosatetraenoic acids (HETE), hydroxyeicosatrienoic acid (HETrE), and PUFA compounds, while the commercially available Strata-X showed a higher recovery for dihydroxyeicosatetraenoic acid (diHETrEs). In summary, our results suggest that this new material can be successfully implemented for the extraction of lipid mediators from biological samples.