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Integrating the summary statistics from a genome-wide association study and expression quantitative trait loci data provides a powerful way of identifying genes with expression levels that are potentially associated with complex diseases. We introduce a parameter called T-score that quantifies the genetic overlap between a gene and the disease phenotype based on the summary statistics, based on the mean values of two Gaussian sequences. Specifically, given two independent samples xn ~ N(θ;∑1) and yn ~ N(µ;∑2), the T-score is defined as ∑i=1n /θiµi/, a nonsmooth functional, that characterizes the number of shared signals between two absolute normal mean vectors /θ/ and /µ/. Using approximation theory, esti- mators are constructed and shown to be minimax rate-optimal and adaptive over various parameter spaces. Simulation studies demonstrate the superiority of the proposed estimators over existing methods. Lastly, the method is applied to an integrative analysis of heart failure genomics data sets and we identify several genes and biological pathways that are potentially causal to human heart failure. [ABSTRACT FROM AUTHOR]

The 2D localization inverse problem of mathematical geophysics is considered. It is of kind of so-called gravimetric problems, due to the underlying process of primary data gathering, the gravimetry. The latter is systematically fulfilled measuring of the gravitational field on the Earth's surface with special high-precision digital appliances, gravimeters. By virtue of the known Poisson equation, which ties together the gravitational potential and material density, the measured fluctuations of potential gravitational field might point on the increase or decrease of material density under Earth's surface, therefore, indicating, for example, desired mineral fields, or underground voids, filled with liquid or gas (which is not always desired, and even can be dangerous). Here we have to accomplish a coordinate recovery for homogeneous gravitational anomaly by the results of gravimetry. In this work we show that minimization process here isn't trivial: the functional has a derivative in any direction, but its classic gradient does not exist. So, then we should switch on non-gradient methods instead. For the numerical solution of the problem, we applied subsequently the subgradient, Nelder-Mead, and the genetic algorithms. Based on results of these computations, a comparative analysis of the applied algorithms is conducted. Also, here we propose corresponding subgradient esteems and the ready-to-use subgradient formulae for numerical solving of this problem. [ABSTRACT FROM AUTHOR]

In the article we have obtained some estimates of the rate of convergence for the recently proposed by Yu. E.Nesterov method of minimization of a convex Lipschitz-continuous function of two variables on a square with a fixed side. The idea of the method is to divide the square into smaller parts and gradually remove them so that in the remaining sufficiently small part. The method consists in solving auxiliary problems of one-dimensional minimization along the separating segments and does not imply the calculation of the exact value of the gradient of the objective functional. The main result of the paper is proved in the class of smooth convex functions having a Lipschitz-continuous gradient. Moreover, it is noted that the property of Lipschitzcontinuity for gradient is sufficient to require not on the whole square, but only on some segments. It is shown that the method can work in the presence of errors in solving auxiliary one-dimensional problems, as well as in calculating the direction of gradients. Also we describe the situation when it is possible to neglect or reduce the time spent on solving auxiliary one-dimensional problems. For some examples, experiments have demonstrated that the method can work effectively on some classes of non-smooth functions. In this case, an example of a simple non-smooth function is constructed, for which, if the subgradient is chosen incorrectly, even if the auxiliary one-dimensional problem is exactly solved, the convergence property of the method may not hold. Experiments have shown that the method under consideration can achieve the desired accuracy of solving the problem in less time than the other methods (gradient descent and ellipsoid method) considered. Partially, it is noted that with an increase in the accuracy of the desired solution, the operating time for the Yu. E. Nesterovs method can grow slower than the time of the ellipsoid method.

In this paper, based on the fixed point index theory for a class of $\widetilde{J}$-multivalued maps on absolute neighbourhood retracts, we introduce the notion of index of solvability for a variational inequality on a Riemannian manifold involving a multivalued vector field. We describe the main properties of this topological characteristic and use it to justify the existence of a solution for a variational inequality problem. As application, the problem of optimization of a non-smooth functional on a Hadamard manifold is considered. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we obtain the existence of a nontrivial solution for a class of singular quasilinear elliptic equations with critical exponents. The proofs rely on a non-smooth critical point theory, and some techniques used by Brezis and Nirenberg. [Copyright &y& Elsevier]

Let X be a Banach space on which a symmetry group G linearly acts and let J be a G-invariant functional defined on X. In 1979, R. Palais (Comm. Math. Phys. 69 (1979) 19) gave some sufficient conditions to guarantee the so-called “Principle of Symmetric Criticality”: every critical point of J restricted on the subspace of G-symmetric points becomes also a critical point of J on the whole space X. This principle is generalized to the case where J is not differentiable within the setting which does not require the full variational structure under the hypothesis that the action of G is isometry or G is compact. [Copyright &y& Elsevier]