Let $$\mathcal{H}^{1,1} (T^1 )$$ denote the Hardy space of real-valued functions on the unit circle with weak derivatives in the usual real Hardy space $$\mathcal{H}^1 (T^1 )$$ . It is shown that when the weak derivative of a locally Lipschitz continuous function$f$has bounded variation on compact sets the Nemytskii operator$F$, defined by$F(u)=f·u$, maps $$\mathcal{H}^{1,1} (T^1 )$$ continuously into itself. A further condition sufficient for the continuous Fréchet differentiability of$F$is then added.

Numerical encoding plays an important role in DNA sequence analysis via computational methods, in which numerical values are associated with corresponding symbolic characters. After numerical representation, digital signal processing methods can be exploited to analyze DNA sequences. To reflect the biological properties of the original sequence, it is vital that the representation is one-to-one. Chaos Game Representation (CGR) is an iterative mapping technique that assigns each nucleotide in a DNA sequence to a respective position on the plane that allows the depiction of the DNA sequence in the form of image. Using CGR, a biological sequence can be transformed one-to-one to a numerical sequence that preserves the main features of the original sequence. In this research, we propose to encode DNA sequences by considering 2D CGR coordinates as complex numbers, and apply digital signal processing methods to analyze their evolutionary relationship. Computational experiments indicate that this approach gives comparable results to the state-of-the-art multiple sequence alignment method, Clustal Omega, and is significantly faster. The MATLAB code for our method can be accessed from: www.mathworks.com/matlabcentral/fileexchange/57152

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We revisit some properties of AdS2 Einstein-Maxwell gravity with the aim of reconciling apparently conflicting results in prior literature. We show that the two dimensional theory can be obtained as a dimensional reduction of the three dimensional Einstein gravity with the CSS boundary condition. In general, this theory on AdS2 can be viewed as an effective theory of the near horizon geometry of a (near-)extremal black hole. We provide an interpretation of our results in terms of the microscopic description of an extremal black hole.

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