Positive Real Method | ScienceGate

Abstract Background Coronavirus disease (COVID-19) presents in children usually with less severe manifestations than in adults. Although fever and cough were reported as the most common symptoms, children can have non-specific symptoms. We describe an infant with aplastic anemia as the main manifestation. Case presentation We describe a case of SARS-CoV-2 infection in an infant without any respiratory symptoms or signs while manifesting principally with pallor and purpura. Pancytopenia with reticulocytopenia was the predominant feature in the initial laboratory investigations, pointing to aplastic anemia. Chest computed tomography surprisingly showed typical findings suggestive of SARS-CoV-2 infection. Infection was later confirmed by positive real-time reverse transcription polymerase chain reaction assay (RT-PCR) for SARS-CoV-2. Conclusions Infants with COVID-19 can have non-specific manifestations and a high index of suspicion should be kept in mind especially in regions with a high incidence of the disease. Chest computed tomography (CT) and testing for SARS-CoV-2 infection by RT-PCR may be considered even in the absence of respiratory manifestations.

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Multiple solutions for critical Choquard-Kirchhoff type equations

Advances in Nonlinear Analysis ◽

10.1515/anona-2020-0119 ◽

2020 ◽

Vol 10 (1) ◽

pp. 400-419 ◽

Cited By ~ 1

Author(s):

Sihua Liang ◽

Patrizia Pucci ◽

Binlin Zhang

Keyword(s):

Critical Exponent ◽

Critical Exponents ◽

Sobolev Inequality ◽

Multiple Solutions ◽

Concentration Compactness ◽

Positive Real ◽

Multiplicity Results ◽

Kirchhoff Type ◽

Concentration Compactness Principle ◽

Compactness Principle

Abstract In this article, we investigate multiplicity results for Choquard-Kirchhoff type equations, with Hardy-Littlewood-Sobolev critical exponents, $$\begin{array}{} \displaystyle -\left(a + b\int\limits_{\mathbb{R}^N} |\nabla u|^2 dx\right){\it\Delta} u = \alpha k(x)|u|^{q-2}u + \beta\left(\,\,\displaystyle\int\limits_{\mathbb{R}^N}\frac{|u(y)|^{2^*_{\mu}}}{|x-y|^{\mu}}dy\right)|u|^{2^*_{\mu}-2}u, \quad x \in \mathbb{R}^N, \end{array}$$ where a > 0, b ≥ 0, 0 < μ < N, N ≥ 3, α and β are positive real parameters, $\begin{array}{} 2^*_{\mu} = (2N-\mu)/(N-2) \end{array}$ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality, k ∈ Lr(ℝN), with r = 2∗/(2∗ − q) if 1 < q < 2* and r = ∞ if q ≥ 2∗. According to the different range of q, we discuss the multiplicity of solutions to the above equation, using variational methods under suitable conditions. In order to overcome the lack of compactness, we appeal to the concentration compactness principle in the Choquard-type setting.

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Analytic functions possessing a positive real part

Duke Mathematical Journal ◽

10.1215/s0012-7094-48-01592-0 ◽

1948 ◽

Vol 15 (4) ◽

pp. 1033-1042 ◽

Cited By ~ 2

Author(s):

Zeev Nehari

Keyword(s):

Analytic Functions ◽

Positive Real Part ◽

Positive Real

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Initial logarithmic coefficients for functions starlike with respect to symmetric points

Boletín de la Sociedad Matemática Mexicana ◽

10.1007/s40590-021-00370-y ◽

2021 ◽

Vol 27 (3) ◽

Author(s):

Paweł Zaprawa

Keyword(s):

Exact Result ◽

Positive Real Part ◽

Additional Benefit ◽

Extremal Functions ◽

Positive Real ◽

Logarithmic Coefficients ◽

Symmetric Points

AbstractIn this paper, we obtain the bounds of the initial logarithmic coefficients for functions in the classes $${\mathcal {S}}_S^*$$ S S ∗ and $${\mathcal {K}}_S$$ K S of functions which are starlike with respect to symmetric points and convex with respect to symmetric points, respectively. In our research, we use a different approach than the usual one in which the coeffcients of f are expressed by the corresponding coeffcients of functions with positive real part. In what follows, we express the coeffcients of f in $${\mathcal {S}}_S^*$$ S S ∗ and $${\mathcal {K}}_S$$ K S by the corresponding coeffcients of Schwarz functions. In the proofs, we apply some inequalities for these functions obtained by Prokhorov and Szynal, by Carlson and by Efraimidis. This approach offers a additional benefit. In many cases, it is easily possible to predict the exact result and to select extremal functions. It is the case for $${\mathcal {S}}_S^*$$ S S ∗ and $${\mathcal {K}}_S$$ K S .

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Eisenstein series and an asymptotic for the K-Bessel function

The Ramanujan Journal ◽

10.1007/s11139-020-00358-8 ◽

2021 ◽

Author(s):

Jimmy Tseng

Keyword(s):

Asymptotic Expansion ◽

Bessel Function ◽

Eisenstein Series ◽

Uniform Estimate ◽

Positive Real ◽

Dominant Term ◽

Complex Order ◽

Fixed Parameter ◽

Real Argument ◽

Real Analytic

AbstractWe produce an estimate for the K-Bessel function $$K_{r + i t}(y)$$ K r + i t ( y ) with positive, real argument y and of large complex order $$r+it$$ r + i t where r is bounded and $$t = y \sin \theta $$ t = y sin θ for a fixed parameter $$0\le \theta \le \pi /2$$ 0 ≤ θ ≤ π / 2 or $$t= y \cosh \mu $$ t = y cosh μ for a fixed parameter $$\mu >0$$ μ > 0 . In particular, we compute the dominant term of the asymptotic expansion of $$K_{r + i t}(y)$$ K r + i t ( y ) as $$y \rightarrow \infty $$ y → ∞ . When t and y are close (or equal), we also give a uniform estimate. As an application of these estimates, we give bounds on the weight-zero (real-analytic) Eisenstein series $$E_0^{(j)}(z, r+it)$$ E 0 ( j ) ( z , r + i t ) for each inequivalent cusp $$\kappa _j$$ κ j when $$1/2 \le r \le 3/2$$ 1 / 2 ≤ r ≤ 3 / 2 .

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