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We consider a inhomogeneous semilinear wave equation on a noncompact complete Riemannian manifold (Formula presented.) of dimension (Formula presented.), without boundary. The reaction exhibits the combined effects of a critical term and of a forcing term. Using a rescaled test function argument together with appropriate estimates, we show that the equation admits no global solution. Moreover, in the special case when (Formula presented.), our result improves the existing literature. Namely, our main result is valid without assuming that the initial values are compactly supported.

Patients with coronavirus disease 2019 (COVID-19) exhibit a wide clinical spectrum ranging from mild respiratory symptoms to critical and fatal diseases, and older individuals are known to be more severely affected. The underlying mechanism of this phenomenon is unknown. A neutralizing antibody against viruses is known to be important to eliminate the virus. In addition, this antibody is induced at high levels in patients with severe COVID-19, followed by a termination of virus replication. Severe COVID-19 patients exhibit high levels of cytokines/chemokines, even after the disappearance of the virus. This indicates that cytokines/chemokines play significant roles in disease severity. These findings also suggest that antiviral therapy (monoclonal antibody and/or convalescent plasma therapy) should be administered early to eliminate the virus, followed by steroid treatment after viral genome disappearance, especially in patients with severe symptoms.

BACKGROUND Coronavirus disease 2019 (COVID-19) severity is classified as asymptomatic, mild, moderate, severe, and critical. Mild cases account for a large percentage of cases in the epidemic and typically exhibit a favorable prognosis. However, a 49%-67% mortality is noted in critical cases. No COVID-19-specific drug has been reported to date, and symptomatic and optimal supportive care, including oxygenation, anti-coinfection treatments, and ventilation, represent the mainstay of treatment for this disease, especially in critical patients. CASE SUMMARY In the above-mentioned context, we share our experience with the treatment of one critical COVID-19 case and review the relevant literature. CONCLUSION Timely tracheal intubation, reasonable mechanical ventilation support, appropriate anti-infection treatment, and early anticoagulation and immunity support are key factors in the successful treatment of this case.

Supplemental Digital Content is available in the text., Background: The outbreak of coronavirus disease 2019 (COVID-19) is becoming a global threat. However, our understanding of the clinical characteristics and treatment of critically ill pediatric patients and their ability of transmitting the coronavirus that causes COVID-19 still remains inadequate because only a handful pediatric cases of COVID-19 have been reported. Methods: Epidemiology, clinical characteristics, treatment, laboratory data and follow-up information and the treatment of critically ill infant were recorded. Results: The infant had life-threatening clinical features including high fever, septic shock, recurrent apnea, petechiae and acute kidney injury and persistent declined CD3+, CD4+ and CD8+ T cells. The duration of nasopharyngeal virus shedding lasted for 49 days even with the administration of lopinavir/ritonavir for 8 days. The CD3+, CD4+ and CD8+ T cells was partially recovered 68 days post onset of the disease. Accumulating of effector memory CD4+ T cells (CD4+TEM) was observed among T-cell compartment. The nucleic acid tests and serum antibody for the severe acute respiratory syndrome coronavirus 2 of the infant’s mother who kept intimate contact with the infant were negative despite no strict personal protection. Conclusions: The persistent reduction of CD4+ and CD8+ T cells was the typical feature of critically ill infant with COVID-19. CD4+ and CD8+ T cells might play a key role in aggravating COVID-19 and predicts a more critical course in children. The prolonged nasopharyngeal virus shedding was related with the severity of respiratory injury. The transmission of SARS-CoV-2 from infant (even very critical cases) to adult might be unlikely.

Purpose Covid 19 pandemic has placed challenges in front of medical health sciences in terms of management, prevention and immunity building. Effectiveness of any medication has not conclusively proven; hence there is need for integrative management of covid 19. We have managed a critical case of covid-19 having history of thalassemia, hypothyroidism with integrative management of Ayurveda and modern medicine. Method A 59 years’ male patient with history of thalassemia had complaints of cough and breathlessness since 4 days. He performed RT PCR because he came in close contact with his daughters who were Covid positive. He was treated with Favipiravir at home for 5 days. He deteriorated on 6th day with SPO2 dropped to 75%, temp raised to 101 F and respiratory rate (RR) raised to 45/min. He was admitted in Yogeshwari Hospital Daund, Maharashtra; treated with oxygen inhalation, Remdesvir and Ayurveda medicines in intensive care unit (ICU). Ayurveda treatment protocol was advised through telemedicine. Result Significant improvement in clinical symptoms was observed and normal HRCT was observed after complete treatment. Conclusion This case report gives us further directions for integrative management in cases of Covid 19. Further clinical researches in this direction are warranted.

Let (Z n) n≥0 with Z n = (Z n (i, j)) 1≤i,j≤p be a p multi-type critical branching process in random environment, and let M n be the expectation of Z n given a fixed environment. We prove theorems on convergence in distribution of sequences of branching processes Zn |Mn| /|Z n | > 0 and ln Zn √ n /|Z n | > 0. These theorems extend similar results for single-type critical branching process in random environment.; Soit (Z n) n≥0 , avec Z n = (Z n (i, j)) 1≤i,j≤p, un processus de branchement multi-type et critique en environnement aléatoire. Soit M n la matrice moyenne de Z n à environnement fixé. Nous démontrons des théorèmes de convergence en distribution des suites (Zn/ |Mn| ) et ( Zn /√ n) conditionnellement à la non extinction (|Z n | > 0). Ces théorèmes étendent dans le cadre multi-type des résultats antérieurs connus pour des processus uni-type.

Let (Z n) n≥0 with Z n = (Z n (i, j)) 1≤i,j≤p be a p multi-type critical branching process in random environment, and let M n be the expectation of Z n given a fixed environment. We prove theorems on convergence in distribution of sequences of branching processes Zn |Mn| /|Z n | > 0 and ln Zn √ n /|Z n | > 0. These theorems extend similar results for single-type critical branching process in random environment.; Soit (Z n) n≥0 , avec Z n = (Z n (i, j)) 1≤i,j≤p, un processus de branchement multi-type et critique en environnement aléatoire. Soit M n la matrice moyenne de Z n à environnement fixé. Nous démontrons des théorèmes de convergence en distribution des suites (Zn/ |Mn| ) et ( Zn /√ n) conditionnellement à la non extinction (|Z n | > 0). Ces théorèmes étendent dans le cadre multi-type des résultats antérieurs connus pour des processus uni-type.

We establish weak well-posedness for critical symmetric stable driven SDEs in R d with additive noise Z, d $\ge$ 1. Namely, we study the case where the stable index of the driving process Z is $\alpha$ = 1 which exactly corresponds to the order of the drift term having the coefficient b which is continuous and bounded. In particular, we cover the cylindrical case when Zt = (Z 1 t ,. .. , Z d t) and Z 1 ,. .. , Z d are independent one dimensional Cauchy processes. Our approach relies on L p-estimates for stable operators and uses perturbative arguments. 1. Statement of the problem and main results We are interested in proving well-posedness for the martingale problem associated with the following SDE: (1.1) X t = x + t 0 b(X s)ds + Z t , where (Z s) s$\ge$0 stands for a symmetric d-dimensional stable process of order $\alpha$ = 1 defined on some filtered probability space ($\Omega$, F, (F t) t$\ge$0 , P) (cf. [2] and the references therein) under the sole assumptions of continuity and boundedness on the vector valued coefficient b: (C) The drift b : R d $\rightarrow$ R d is continuous and bounded. 1 Above, the generator L of Z writes: L$\Phi$(x) = p.v. R d \{0} [$\Phi$(x + z) -- $\Phi$(x)]$\nu$(dz), x $\in$ R d , $\Phi$ $\in$ C 2 b (R d), $\nu$(dz) = d$\rho$ $\rho$ 2$\mu$ (d$\theta$), z = $\rho$$\theta$, ($\rho$, $\theta$) $\in$ R * + x S d--1. (1.2) (here $\times$, $\times$ (or $\times$) and | $\times$ | denote respectively the inner product and the norm in R d). In the above equation, $\nu$ is the L{\'e}vy intensity measure of Z, S d--1 is the unit sphere of R d and$\mu$ is a spherical measure on S d--1. It is well know, see e.g. [20] that the L{\'e}vy exponent $\Phi$ of Z writes as: (1.3) $\Phi$($\lambda$) = E[exp(i $\lambda$, Z 1)] = exp -- S d--1 | $\lambda$, $\theta$ |$\mu$(d$\theta$) , $\lambda$ $\in$ R d , where $\mu$ = c 1$\mu$ , for a positive constant c 1 , is the so-called spectral measure of Z. We will assume some non-degeneracy conditions on $\mu$. Namely we introduce assumption (ND) There exists $\kappa$ $\ge$ 1 s.t. (1.4) $\forall$$\lambda$ $\in$ R d , $\kappa$ --1 |$\lambda$| $\le$ S d--1 | $\lambda$, $\theta$ |$\mu$(d$\theta$) $\le$ $\kappa$|$\lambda$|. 1 The boundedness of b is here assumed for technical simplicity. Our methodology could apply, up to suitable localization arguments, to a drift b having linear growth.

In this paper we present a short description of the PhD thesis with the title ”Applications of Hamiltonian systems in analysis and optimization”. This thesis was defended on November 2-nd 2017 at the Institute of Mathematics ”Simion Stoilow” of the Romanian Academy.

Under some conditions, an asymptotic solution containing boundary functions was constructed in a paper by Vasil&rsquo, eva and Butuzov (Differ. Uravn. 1970, 6(4), 650&ndash, 664 (in Russian), English transl.: Differential Equations 1971, 6, 499&ndash, 510) for an initial value problem for weakly non-linear differential equations with a small parameter standing before the derivative, in the case of a singular matrix A ( t ) standing in front of the unknown function. In the present paper, the orthogonal projectors onto k e r A ( t ) and k e r A ( t ) &prime, (the prime denotes the transposition) are used for asymptotics construction. This approach essentially simplifies understanding of the algorithm of asymptotics construction.

In [8], the existence of the solution is proved for a scalar linearly growingbackward stochastic differential equation (BSDE) when the terminal value is$L\exp (\mu \sqrt{2\log (1+L)} )$-integrable for a positive parameter $\mu >\mu _{0}$ with a critical value $\mu _{0}$, and a counterexample is provided to show that the preceding integrability for $\mu \mu _{0}$) is also given in [3] for the preceding BSDE under the uniformly Lipschitz condition of the generator. In this note, we prove that these two results still hold for the critical case: $\mu =\mu _{0}$.

The theory of multi-type branching process in i.i.d. environment is considerably less developed than for the univariate case, and fundamental questions are up to date unsolved. Answers demand a deep understanding of the behaviour of products of i.i.d. matrices with non-negative entries.Under mild assumptions, when the probability generating functions of the reproduction laws are fractional-linear, the survival probability of the multi-type branching process in random environment up to moment n is proportional to 1/√n as n → ∞.Techniques for univariate branching processes in random environment and methods from the theory of products of i.i.d. random matrices are required.; La théorie des processus de branchement multi-type en environnement i.i.d. est considérablement moins développée que dans le cas univarié, et les questions fondamentales ne sont pas résolues en totalité à ce jour. Les réponses exigent une compréhension profonde du comportement des produits des matrices i.i.d. \`a coefficients positifs.Sous des hypothèses assez générales et lorsque les fonctions génératrices de probabilité des lois de reproduction sont "linéaire fractionnaires'', nous montrons que la probabilité de survie à l'instant n du processus de branchement multi-type en environnement aléatoire est proportionnelle 1/√n lorsque n → ∞.La déemonstration de ce réesultat suit l'approche développée pour étudier les processus de branchement uni-variés en environnement aléatoire i. i. d. Il utilise de façon cruciale des résultats récents portant sur les fluctuations des normes de produits de matrices aléatoires i.i.d.

We discuss the solution of implicit systems in the critical case, i.e. when the classical assumptions of the implicit functions theorem are not satised. The generalized solution introduced bellow solves such cases and it may not be a manifold. In certain examples, it may have a complex structure and its approximation is nontrivial. We present here an algorithm for the approximation of the generalized solution. Numerical tests are also included.

National Research Center for Maternal and Child Health, part of corporate fund “University Medical Center” (NRCMC, CP “UMC”) conducted a monitoring and analysis of critical cases in obstetrics in 2015-2017. The purpose of the monitoring is to decrease the maternal deaths. The analysis was prepared based on the data presented by the regions. The methods of research are analysis of record forms of pregnant women, parturient women and new mothers in critical condition.

Conditioned on the generating functions of offspring distribution, we study the asymp-totic behaviour of the probability of non-extinction of a critical multi-type Galton-Watson process in i.i.d. random environments by using limits theorems for products of positive random matrices. Under some certain assumptions, the survival probability is proportional to 1/ √ n.

The theory of multi-type branching process in i.i.d. environment is considerably less developed than for the univariate case, and fundamental questions are up to date unsolved. Answers demand a deep understanding of the behaviour of products of i.i.d. matrices with non-negative entries.Under mild assumptions, when the probability generating functions of the reproduction laws are fractional-linear, the survival probability of the multi-type branching process in random environment up to moment n is proportional to 1/√n as n → ∞.Techniques for univariate branching processes in random environment and methods from the theory of products of i.i.d. random matrices are required., La théorie des processus de branchement multi-type en environnement i.i.d. est considérablement moins développée que dans le cas univarié, et les questions fondamentales ne sont pas résolues en totalité à ce jour. Les réponses exigent une compréhension profonde du comportement des produits des matrices i.i.d. \`a coefficients positifs.Sous des hypothèses assez générales et lorsque les fonctions génératrices de probabilité des lois de reproduction sont "linéaire fractionnaires'', nous montrons que la probabilité de survie à l'instant n du processus de branchement multi-type en environnement aléatoire est proportionnelle 1/√n lorsque n → ∞.La déemonstration de ce réesultat suit l'approche développée pour étudier les processus de branchement uni-variés en environnement aléatoire i. i. d. Il utilise de façon cruciale des résultats récents portant sur les fluctuations des normes de produits de matrices aléatoires i.i.d.

Special issue; International audience; In [3], the authors proved that uniqueness holds among solutions whose exponentials are $L^p$ with $p$ bigger than a constant $\gamma$ ($p>\gamma$). In this paper, we consider the critical case: $p=\gamma$. We prove that the uniqueness holds among solutions whose exponentials are $L^\gamma$ under the additional assumption that the generator is strongly convex.

This study aims to describe how researchers may incorporate four different generalisation methods (working hypothesis, analytic generalisation, critical case and naturalistic generalisation) to aid them in properly generaliseing from case study results. It was found that critical cases can be used in combination with working hypothesis to determine the falsifiability of the hypotheses. Working hypothesis and analytic generalisation complement each other by making it possible for researchers to generalise to both theory and new hypotheses (i.e. both inductively and deductively). Lastly, naturalistic and analytic generalisation enables a double-ended generalisation, where both the reader and the scientist themself generalise to specific situations and over-all theory, respectively.

The periodic quasilinear system of differential equations with small parameter and piecewise constant argument of generalized type [M.U. Akhmet, Integral manifolds of differential equations with piecewise constant argument of generalized type, Nonlinear Anal. TMA, 66 (2007) 367-383, M.U. Akhmet, On the reduction principle for differential equations with piecewise argument of generalized type, J. Math. Anal. Appl. 336 (2007) 646-663] is addressed. We consider the critical case, when associated linear homogeneous system admits nontrivial periodic solutions. Criteria of existence of periodic solutions of such equations are obtained. One of the main auxiliary results of our paper is an analogue of Gronwall-Bellman Lemma for functions with piecewise constant and retarded-advanced type arguments. Dependence of solutions on the parameter is investigated. Appropriate examples are given to show our results. (c) 2008 Elsevier Ltd. All rights reserved.

In this paper, we study the conditions under which the zero solution is stable in the semi- liner case for certain third order differential equation of the form : Where , s = 1,2,3 , , , The characteristic equation of the above differential equation has complex roots of the form : and the other root has the following , . property

The stability with respect to a part of variables in the critical case of pair of pure imaginary and single zero roots is researched. Also case of double zero roots with two solution groups and case of two pairs of pure imaginary roots are considered. Particular conditions of the partial asymptotic stability groups) and the partial instability were obtained. In conclusion, we give an example of solving the problem. Исследуется устойчивость относительно части переменных в критическом случае пары чисто мнимых и одного нулевого корней. Также рассмотрены случай двойного нулевого корня с двумя группами решений и случай двух пар чисто мнимых корней. Для данных случаев получены частные условия асимптотической устойчивости по части переменных и неустойчивости. В заключение приведен пример решения задачи.

Исследуется устойчивость относительно части переменных в критическом случае двойного нулевого корня (m=2). Рассмотрены случаи двойного нулевого корня с одной и с двумя группами решений. Для данных случаев получены частные условия асимптотической устойчивости по части переменных (при двойном нулевом корне с двумя группами решений) и неустойчивости (при двойном нулевом корне с одной группой решений). The stability with respect to a part of variables in the critical case of double zero roots (m=2) is investigated. Cases of double zero roots with one solution group or two solution groups are considered. Particular conditions of the partial asymptotic stability (case of two solution groups) and the partial instability (case of one solution group) were obtained.

In this paper, we consider a kind of neutral functional differential equation as follows: x(t)−cx(t−τ)′=gt,xt−μ(t)+e(t) in the critical case |c|=1. By employing the topological degree theory and some analysis techniques, we obtain some new results on the existence of periodic solutions.

Рассмотрены слабонелинейные краевые задачи для операторных уравнений с нетеровым оператором в линейной части краевой задачи в критическом случае. Получены необходимые и достаточные условия существования единственного решения, построена сходящаяся итерационная процедура для его построения., The paper highlights the weak non-linear boundary value problems for operator equations with Noether's operator in the linear part of the boundary value problem in critical cases. The author has obtained necessary and sufficient conditions for finding the only possible solutions of such boundary value problem. The author also managed to establish the converging iteration procedure for finding a solution is proposed., Розглянуто слабонелінійні крайові задачі для операторних рівнянь з нетеровим оператором в лінійній частині крайової задачі у критичному випадку. Отримані необхідні та достатні умови існування єдиного розв'язку, побудовано збіжну ітераційну процедуру для його побудови.

The strength and stability properties of hierarchical load bearing networks and their strengthened variants have been discussed in recent work. Here, we study the avalanche time distributions on these load bearing networks. The avalanche time distributions of the V- lattice, a unique realization of the networks, show power-law behavior when tested with certain fractions of its trunk weights. All other avalanche distributions show Gaussian peaked behavior. Thus the V- lattice is the critical case of the network. We discuss the implications of this result., Comment: 7 pages, 10 figures, To appear in the Proceedings of "Perspectives in Nonlinear Dynamics 2010", Accepted in a special issue of Pramana - journal of physics

We investigate the statistics and dynamics of failure in a two-dimensional load-bearing network with branching hierarchical structure, and its variants. The variants strengthen the original lattice by using connectivity strategies which add new sites to the maximal cluster in top-to-bottom or bottom-to-top versions. We study the load-bearing capacity and the failure tolerance of all versions, as well as that of the strongest realization of the original lattice, the V lattice. The average number of failures as a function of the test load shows power-law behavior with power 5/2 for the V lattice, but sigmoidal behavior for all other versions. Thus the V lattice turns out to be the critical case of the load-bearing lattices. The distribution of failures is Gaussian for the original lattice, the V lattice, and the bottom-to-top strategy, but is non-Gaussian for the top-to-bottom one. The bottom-to-top strategy leads to stable and strong lattices, and can resist failure even when tested with weights which greatly exceed the capacity of its backbone. We also examine the behavior of asymmetric lattices and discover that the mean failure rates are minimized if the probability of connection p is symmetric with respect to both neighbors. Our results can be of relevance in the context of realistic networks. � 2011 American Physical Society.

In this paper, for the “critical case” with two delays, we establish two relations between any two solutions y ( t ) and y ∗ ( t ) for the Volterra integral equation of non-convolution type y ( t ) = f ( t ) + ∫ t − τ t − δ k ( t , s ) g ( y ( s ) ) d s and a solution z ( t ) of the first order differential equation z ( t ) = β ( t ) [ z ( t − δ ) − z ( t − τ ) ] , and offer a sufficient condition that lim t → + ∞ ( y ( t ) − y ∗ ( t ) ) = 0 .

We obtain coefficient conditions of existence and an iterational algorithm for constructing periodic solutions of weakly perturbed autonomous nonlinear differential systems in critical cases in the presence of multiple roots for generating amplitudes., Отримано коефіцієнтні умови існування та ітераційний алгоритм побудови періодичних розв'язків слабозбурених автономних нелінійних диференціальних систем в критичних випадках при кратних коренях рівняння для породжуючих амплітуд.

A scalar linear differential equation with time-dependent delay x@?(t)=-a(t)x(t-@t(t)) is considered, where t@?I@?[t"0,~), t"0@?R, a:I->R^+@?(0,~) is a continuous function and @t:I->R^+ is a continuous function such that t-@t(t)>t"0-@t(t"0) if t>t"0. The goal of our investigation is to give sufficient conditions for the existence of positive solutions as t->~ in the critical case in terms of inequalities on a and @t. A generalization of one known final (in a certain sense) result is given for the case of @t being not a constant. Analysing this generalization, we show, e.g., that it differs from the original statement with a constant delay since it does not give the best possible result. This is demonstrated on a suitable example.