Deposits of clastic carbonate-dominated (calciclastic) sedimentary slope systems in the rock record have been identified mostly as linearly-consistent carbonate apron deposits, even though most ancient clastic carbonate slope deposits fit the submarine fan systems better. Calciclastic submarine fans are consequently rarely described and are poorly understood. Subsequently, very little is known especially in mud-dominated calciclastic submarine fan systems. Presented in this study are a sedimentological core and petrographic characterisation of samples from eleven boreholes from the Lower Carboniferous of Bowland Basin (Northwest England) that reveals a >250 m thick calciturbidite complex deposited in a calciclastic submarine fan setting. Seven facies are recognised from core and thin section characterisation and are grouped into three carbonate turbidite sequences. They include: 1) Calciturbidites, comprising mostly of highto low-density, wavy-laminated bioclast-rich facies; 2) low-density densite mudstones which are characterised by planar laminated and unlaminated muddominated facies; and 3) Calcidebrites which are muddy or hyper-concentrated debrisflow deposits occurring as poorly-sorted, chaotic, mud-supported floatstones. These

Phd by thesis

We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

We study the connection between the highly non-convex loss function of a simple model of the fully-connected feed-forward neural network and the Hamiltonian of the spherical spin-glass model under the assumptions of: i) variable independence, ii) redundancy in network parametrization, and iii) uniformity. These assumptions enable us to explain the complexity of the fully decoupled neural network through the prism of the results from random matrix theory. We show that for large-size decoupled networks the lowest critical values of the random loss function form a layered structure and they are located in a well-defined band lower-bounded by the global minimum. The number of local minima outside that band diminishes exponentially with the size of the network. We empirically verify that the mathematical model exhibits similar behavior as the computer simulations, despite the presence of high dependencies in real networks. We conjecture that both simulated annealing and SGD converge to the band of low critical points, and that all critical points found there are local minima of high quality measured by the test error. This emphasizes a major difference between largeand small-size networks where for the latter poor quality local minima have nonzero probability of being recovered. Finally, we prove that recovering the global minimum becomes harder as the network size increases and that it is in practice irrelevant as global minimum often leads to overfitting.

/pdf/the-loss-surfaces-of-multilayer-networks-5c4ib8rk8y.pdf

The Loss Surfaces of Multilayer Networks

CERN ´ Y Abstract. We give an asymptotic evaluation of the complexity of spherical p-spin spin- glass models via random matrix theory. This study enables us to obtain detailed infor- mation about the bottom of the energy landscape, including the absolute minimum (the ground state), the other local minima, and describe an interesting layered structure of the low critical values for the Hamiltonians of these models. We also show that our ap- proach allows us to compute the related TAP-complexity and extend the results known in the physics literature. As an independent tool, we prove a LDP for the k-th largest eigenvalue of the GOE, extending the results of (BDG01). How many critical values of given index and below a given level does a typical random Morse function have on a high dimensional manifold? Our work addresses this question in a very special case. We look at certain natural random Gaussian functions on the N- dimensional sphere known as p-spin spherical spin glass models. We cannot yet answer the question above about the typical number, but we can study thoroughly the mean number, which we show is exponentially large in N. We introduce a new identity, based on the classical Kac-Rice formula, relating random matrix theory and the problem of counting these critical values. Using this identity and tools from random matrix theory, we give an asymptotic evaluation of the complexity of these spherical spin-glass models. The complexity mentioned here is defined as the mean number of critical points of given index whose value is below (or above) a given level. This includes the important question of counting the mean number of local minima below a given level, and in particular the question of finding the ground state energy (the minimal value of the Hamiltonian). We show that this question is directly related to the study of the edge of the spectrum of the Gaussian Orthogonal Ensemble (GOE). The question of computing the complexity of mean-field spin glass models has recently been thoroughly studied in the physics literature (see for example (CLR03) and the refer- ences therein), mainly for a different measure of the complexity, i.e. the mean number of solutions to the Thouless-Anderson-Palmer equations, or TAP-complexity. Our approach to the complexity enables us to recover known results in the physics literature about TAP-complexity, to compute the ground state energy (when p is even), and to describe an interesting layered structure of the low energy levels of the Hamiltonians of these models, which might prove useful for the study of the metastability of Langevin dynamics for these models (in longer time scales than those studied in (BDG01)). The paper is organised as follows. In Section 2, we give our main results. In Section 3, we prove two main formulas (Theorem 2.1 and 2.2), relating random matrix theory (specif- ically the GOE) and spherical spin glasses. These formulas are direct consequences of the Kac-Rice formula (we learned the version needed here in the book (AT07), for another modern account see (AW09)). The main new ingredient is the fact that, for spherical spin- glass models, the Hessian of the Hamiltonian at a critical point, conditioned on the value of the Hamiltonian, is a symmetric Gaussian random matrix with independent entries (up to symmetry) plus a diagonal matrix. This implies, in particular, that it is possible to

Random matrices and complexity of spin glasses

We give the "quenched" scaling limit of Bouchaud's trap model in d ≥ 2. This scaling limit is the fractional-kinetics process, that is the time change of a d-dimensional Brownian motion by the inverse of an independent α-stable subordinator.

/pdf/scaling-limit-for-trap-models-on-d-gw7sjq43oj.pdf

Scaling limit for trap models on ℤd

We give a general proof of aging for trap models using the arcsine law for stable subordinators. This proof is based on abstract conditions on the potential theory of the underlying graph and on the randomness of the trapping landscape. We apply this proof to aging for trap models on large, two-dimensional tori and for trap dynamics of the random energy model on a broad range of time scales. (C) 2006 Wiley Periodicals, Inc.

The arcsine law as a universal aging scheme for trap models

We consider a random walk among unbounded random conductances whose distribution has infinite expectation and polynomial tail. We prove that the scaling limit of this process is a Fractional-Kinetics process—that is the time change of a d-dimensional Brownian motion by the inverse of an independent α-stable subordinator. We further show that the same process appears in the scaling limit of the non-symmetric Bouchaud’s trap model.

/pdf/convergence-to-fractional-kinetics-for-random-walks-4abp22x6va.pdf

Convergence to fractional kinetics for random walks associated with unbounded conductances

Let E x be a collection of i.i.d. exponential random variables. Symmetric Bouchaud's model on ℤ2 is a Markov chain X(t) whose transition rates are given by w xy = ν exp (−βE x ) if x, y are neighbours in ℤ2. We study the behaviour of two correlation functions: ℙ[X(t w +t) = X(t w )] and ℙ[X(t') = X(t w ) ∀ t'∈ [t w , t w + t]]. We prove the (sub)aging behaviour of these functions when β > 1.

/pdf/aging-in-two-dimensional-bouchaud-s-model-40b265avlw.pdf

Jiří Černý

Papers

Random matrices and complexity of spin glasses

Scaling limit for trap models on ℤd

The arcsine law as a universal aging scheme for trap models

Convergence to fractional kinetics for random walks associated with unbounded conductances

Aging in two-dimensional Bouchaud's model