We prove that uniform random quadrangulations of the sphere with$n$faces, endowed with the usual graph distance and renormalized by$n$^{−1/4}, converge as$n$→$∞$in distribution for the Gromov–Hausdorff topology to a limiting metric space. We validate a conjecture by Le Gall, by showing that the limit is (up to a scale constant) the so-called$Brownian map$, which was introduced by Marckert–Mokkadem and Le Gall as the most natural candidate for the scaling limit of many models of random plane maps. The proof relies strongly on the concept of$geodesic stars$in the map, which are configurations made of several geodesics that only share a common endpoint and do not meet elsewhere.

We study different classes of Jensen measures for plurisubharmonic functions, in particular the relation between Jensen measures for continuous functions and Jensen measures for upper bounded functions. We prove an approximation theorem for plurisubharmonic functions in$B$-regular domain. This theorem implies that the two classes of Jensen measures coincide in$B$-regular domains. Conversely we show that if Jensen measures for continuous functions are the same as Jensen measures for upper bounded functions and the domain is hyperconvex, the domain satisfies the same approximation theorem as above.

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We count the number$S$($x$) of quadruples $ {\left( {x_{1} ,x_{2} ,x_{3} ,x_{4} } \right)} \in \mathbb{Z}^{4} $ for which $$ p = x^{2}_{1} + x^{2}_{2} + x^{2}_{3} + x^{2}_{4} \leqslant x $$ is a prime number and satisfying the determinant condition:$x$_{1}$x$_{4}−$x$_{2}$x$_{3}= 1. By means of the sieve, one shows easily the upper bound$S$($x$) ≪$x$/log$x$. Under a hypothesis about prime numbers, which is stronger than the Bombieri–Vinogradov theorem but is weaker than the Elliott–Halberstam conjecture, we prove that this order is correct, that is$S$($x$) ≫$x$/log$x$.

We study the minimization problem of a non-convex sparsity promoting penalty func- tion, the transformed l1 (TL1), and its application in compressed sensing (CS). The TL1 penalty inter- polates l0 and l1 norms through a nonnegative parameter a ∈ (0, +∞), similar to lp with p ∈ (0, 1], and is known to satisfy unbiasedness, sparsity and Lipschitz continuity properties. We first consider the constrained minimization problem, and discuss the exact recovery of l0 norm minimal solution based on the null space property (NSP). We then prove the stable recovery of l0 norm minimal solution if the sensing matrix A satisfies a restricted isometry property (RIP). We formulated a normalized problem to overcome the lack of scaling property of the TL1 penalty function. For a general sensing matrix A, we show that the support set of a local minimizer corresponds to linearly independent columns of A. Next, we present difference of convex algorithms for TL1 (DCATL1) in computing TL1-regularized con- strained and unconstrained problems in CS. The DCATL1 algorithm involves outer and inner loops of iterations, one time matrix inversion, repeated shrinkage operations and matrix-vector multiplications. The inner loop concerns an l1 minimization problem on which we employ the Alternating Direction Method of Multipliers (ADMM). For the unconstrained problem, we prove convergence of DCATL1 to a stationary point satisfying the first order optimality condition. In numerical experiments, we identify the optimal value a = 1, and compare DCATL1 with other CS algorithms on two classes of sensing ma- trices: Gaussian random matrices and over-sampled discrete cosine transform matrices (DCT). Among existing algorithms, the iterated reweighted least squares method based on l1/2 norm is the best in sparse recovery for Gaussian matrices, and the DCA algorithm based on l1 minus l2 penalty is the best for over-sampled DCT matrices. We find that for both classes of sensing matrices, the performance of DCATL1 algorithm (initiated with l1 minimization) always ranks near the top (if not the top), and is the most robust choice insensitive to the conditioning of the sensing matrix A. DCATL1 is also com- petitive in comparison with DCA on other non-convex penalty functions commonly used in statistics with two hyperparameters.