We generate a class of string backgrounds by a sequence of TsT transformations of the NS1-NS5 system that we argue are holographically dual to states in a single-trace TT¯+JT¯+TJ¯-deformed CFT2. The new string backgrounds include general rotating black hole solutions with two U(1) charges as well as a smooth solution without a horizon that is interpreted as the ground state. As evidence for the correspondence we (i) derive the long string spectrum and relate it to the spectrum of the dual field theory; (ii) show that the black hole thermodynamics can be reproduced from single-trace TT¯+JT¯+TJ¯-deformed CFTs; and (iii) show that the energy of the ground state matches the energy of the vacuum in the dual theory. We also study geometric properties of these new spacetimes and find that for some choices of the parameters the three-dimensional Ricci scalar in the Einstein frame can become positive in a region outside the horizon before reaching closed timelike curves and singularities.

Open quantum walks (also known as open quantum random walks) are quantum analogs of classical Markov chains in probability theory, and have potential application in quan- tum information and quantum computation. Quantum Bernoulli noises are annihilation and creation operators acting on Bernoulli functionals, and can be used as the environ- ment of an open quantum system. In this paper, by using quantum Bernoulli noises as the environment, we introduce an open quantum walk on a general higher-dimensional integer lattice. We obtain a quantum channel representation of the walk, which shows that the walk is indeed an open quantum walk. We prove that all the states of the walk are separable provided its initial state is separable. We also prove that, for some initial states, the walk has a limit probability distribution of higher-dimensional Gauss type. Finally we show links between the walk and a unitary quantum walk recently introduced in terms of quantum Bernoulli noises.

Let $f(n)$ be the maximum number of edges in a graph on $n$ vertices in which no two cycles have the same length. P. Erdos raised the problem of determining $f(n)$ (see J.A. Bondy and U.S.R. Murty, Graph Theory with Applications (Macmillan, New York, 1976), p.247, Problem 11). We present the problems, conjectures related to this problems and we summarize the know results. We make the following conjecture: \par \noindent{\bf Conjecture } $$\lim\sb {n \to \infty} {f(n)-n \over \sqrt n} = \sqrt {2 + \frac{4}{9}}.$$

We observe that some self-similar measures defined by finite or infinite iterated function systems with overlaps satisfy certain "bounded measure type condition", which allows us to extract useful measure-theoretic properties of iterates of the measure. We develop a technique to obtain a closed formula for the spectral dimension of the Laplacian defined by self-similar measure satisfying this condition. For Laplacians defined by fractal measures with overlaps, spectral dimension has been obtained earlier only for a small class of one-dimensional self-similar measures satisfying Strichartz second-order self-similar identities. The main technique we use relies on the vector-valued renewal theorem proved by Lau, Wang and Chu.

In this paper, we will prove the Weyl's law for the asymptotic formula of Dirichlet eigenvalues on metric measure spaces with generalized Ricci curvature bounded from below.