Asymptotic safety of quantum gravity beyond Ricci scalars

We investigate the asymptotic safety conjecture for quantum gravity including curvature invariants beyond Ricci scalars. Our strategy is put to work for families of gravitational actions which depend on functions of the Ricci scalar, the Ricci tensor, and products thereof. Combining functional renormalization with high order polynomial approximations and full numerical integration we derive the renormalization group flow for all couplings and analyse their fixed points, scaling exponents, and the fixed point effective action as a function of the background Ricci curvature. The theory is characterized by three relevant couplings. Higher-dimensional couplings show near-Gaussian scaling with increasing canonical mass dimension. We find that Ricci tensor invariants stabilize the UV fixed point and lead to a rapid convergence of polynomial approximations. We apply our results to models for cosmology and establish that the gravitational fixed point admits inflationary solutions. We also compare findings with those from fðRÞ-type theories in the same approximation and pin-point the key new effects due to Ricci tensor interactions. Implications for the asymptotic safety conjecture of gravity are indicated

Deterministic and Probabilistic Boolean Control Networks and their application to Gene Regulatory Networks

This thesis focuses on Deterministic and Probabilistic Boolean Control Networks and their application to some specific Gene Regulatory Networks. At first, some introductory materials about Boolean Logic, Left Semi-tensor Product and Probability are presented in order to explain in detail the concepts of Boolean Networks, Boolean Control Networks, Probabilistic Boolean Networks and Probabilistic Boolean Control Networks. These networks can be modelled in state-space and their representation, obtained by means of the left semi-tensor product, is called algebraic form. Subsequently, the thesis concentrates on presenting the fundamental properties of these networks such as the classical Systems Theory properties of stability, reachability, controllability and stabilisation. Afterwards, the attention is drawn towards the comparison between deterministic and probabilistic boolean networks. Finally, two examples of Gene Regulatory Networks are modelled and analysed by means of a Boolean Network and a Probabilistic Boolean Network.This thesis focuses on Deterministic and Probabilistic Boolean Control Networks and their application to some specific Gene Regulatory Networks. At first, some introductory materials about Boolean Logic, Left Semi-tensor Product and Probability are presented in order to explain in detail the concepts of Boolean Networks, Boolean Control Networks, Probabilistic Boolean Networks and Probabilistic Boolean Control Networks. These networks can be modelled in state-space and their representation, obtained by means of the left semi-tensor product, is called algebraic form. Subsequently, the thesis concentrates on presenting the fundamental properties of these networks such as the classical Systems Theory properties of stability, reachability, controllability and stabilisation. Afterwards, the attention is drawn towards the comparison between deterministic and probabilistic boolean networks. Finally, two examples of Gene Regulatory Networks are modelled and analysed by means of a Boolean Network and a Probabilistic Boolean Network

Inductive and Functional Types in Ludics

Ludics is a logical framework in which types/formulas are modelled by sets of terms with the same computational behaviour. This paper investigates the representation of inductive data types and functional types in ludics. We study their structure following a game semantics approach. Inductive types are interpreted as least fixed points, and we prove an internal completeness result giving an explicit construction for such fixed points. The interactive properties of the ludics interpretation of inductive and functional types are then studied. In particular, we identify which higher-order functions types fail to satisfy type safety, and we give a computational explanation

Strongly convergent approximations to fixed points of total asymptotically nonexpansive mappings

In this work we prove a new strong convergence result of the regularized successive approximation method given by yn+1 = qnz0 + (1 − qn)T n yn, n = 1, 2, ..., where limn→∞ qn = 0 and X∞ n=1 qn = ∞, for T a total asymptotically nonexpansive mapping, i.e., T is such that kT nx − T n yk ≤ kx − yk + k (1) n φ(kx − yk) + k (2) n , where k 1 n and k 2 n are real null convergent sequences and φ : R+ → R+ is continuous and such that φ(0) = 0 and limt→∞ φ(t) t ≤ C for a certain constant C > 0. Among other features, our results essentially generalize existing results on strong convergence for T nonexpansive and asymptotically nonexpansive. The convergence and stability analysis is given for both self- and nonself-mappings

Stability and Convergence Results Based on Fixed Point Theory for a Generalized Viscosity Iterative Scheme

Es reproducción del documento publicado en http://dx.doi.org/10.1155/2009/314581A generalization of Halpern's iteration is investigated on a compact convex subset of a smooth Banach space. The modified iteration process consists of a combination of a viscosity term, an external sequence, and a continuous nondecreasing function of a distance of points of an external sequence, which is not necessarily related to the solution of Halpern's iteration, a contractive mapping, and a nonexpansive one. The sum of the real coefficient sequences of four of the above terms is not required to be unity at each sample but it is assumed to converge asymptotically to unity. Halpern's iteration solution is proven to converge strongly to a unique fixed point of the asymptotically nonexpansive mapping.Ministerio de Educación (DPI2006-00714; Gobierno Vasco (GIC07143-IT-269-07 y SAIOTEK S-PE08UN15

Computation of eigenvalues in proportionally damped viscoelastic structures based on the fixed-point iteration

Linear viscoelastic structures are characterized by dissipative forces that depend on the history of the velocity response via hereditary damping functions. The free motion equation leads to a nonlinear eigenvalue problem characterized by a frequency-dependent damping matrix. In the present paper, a novel and efficient numerical method for the computation of the eigenvalues of linear and proportional or lightly non-proportional viscoelastic structures is developed. The central idea is the construction of two complex-valued functions of a complex variable, whose fixed points are precisely the eigenvalues. This important property allows the use of these functions in a fixed-point iterative scheme. With help of some results in fixed point theory, necessary conditions for global and local convergence are provided. It is demonstrated that the speed of convergence is linear and directly depends on the level of induced damping. In addition, under certain conditions the recursive method can also be used for the calculation of non-viscous real eigenvalues. In order to validate the mathematical results, two numerical examples are analyzed, one for single degree-of-freedom systems and another for multiple ones.The authors gratefully acknowledge the support of the Polytechnic University of Valencia under programs PAID 02-11-1828 and 05-10-2674 and of the National Science and Research Council of Canada.Lázaro Navarro, M.; Pérez Aparicio, JL.; Epstein, M. (2012). Computation of eigenvalues in proportionally damped viscoelastic structures based on the fixed-point iteration. Applied Mathematics and Computation. 219(8):3511-3529. https://doi.org/10.1016/j.amc.2012.09.026S35113529219

Probabilistic Contraction Analysis of Iterated Random Operators

Consider a contraction operator $T$ over a complete metric space $\mathcal X$ with the fixed point $x^\star$. In many computational applications, it is difficult to compute $T(x)$; therefore, one replaces the application contraction operator $T$ at iteration $k$ by a random operator $\hat T^n_k$ using $n$ independent and identically distributed samples of a random variable. Consider the Markov chain $(\hat X^n_k)_{k\in\mathbb{N}}$, which is generated by $\hat X^n_{k+1} = \hat T^n_k(\hat X^n_k)$. In this paper, we identify some sufficient conditions under which (i) the distribution of $\hat X^n_k$ converges to a Dirac mass over $x^\star$ as $k$ and $n$ go to infinity, and (ii) the probability that $\hat X^n_k$ is far from $x^\star$ as $k$ goes to infinity can be made arbitrarily small by an appropriate choice of $n$. We also derive an upper bound on the probability that $\hat X^n_k$ is far from $x^\star$ as $k\rightarrow \infty$. We apply the result to study the convergence in probability of iterates generated by empirical value iteration algorithms for discounted and average cost Markov decision problems.Comment: 37 pages, submitted to SIAM Journal on Control and Optimizatio

Efficient Generic Quotients Using Exact Arithmetic

The usual formulation of efficient division uses Newton iteration to compute an inverse in a related domain where multiplicative inverses exist. On one hand, Newton iteration allows quotients to be calculated using an efficient multiplication method. On the other hand, working in another domain is not always desirable and can lead to a library structure where arithmetic domains are interdependent. This paper uses the concept of a whole shifted inverse and modified Newton iteration to compute quotients efficiently without leaving the original domain. The iteration is generic to domains having a suitable shift operation, such as integers or polynomials with coefficients that do not necessarily commute

Non-reversible stationary states for majority voter and Ising dynamics on trees

We study three Markov processes on infinite, regular trees: the Glauber (heat bath) dynamics of the Ising model, a majority voter dynamic, and a coalescing particle model. In each of the three cases the tree exhibits a preferred direction which is encoded into the model. For all three models, our main result is the existence of a stationary but non-reversible measure. For the Ising model, this requires imposing that the inverse temperature is large and choosing suitable non-uniform couplings, and our theorem implies the existence of a stationary measure which does not at all look like a low-temperature Ising Gibbs measure. The interesting aspect of our results lies in the fact that the analogous processes do not have non-Gibbsian stationary measures on $\mathbb Z^d$, owing to the amenability of that graph. In fact, no example of a stochastic Ising model with a non-reversible stationary state was known to date.Comment: 13 page