There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

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

Random matrix theory has found many applications in physics, statistics and engineering since its inception. Although early developments were motivated by practical experimental problems, random matrices are now used in fields as diverse as Riemann hypothesis, stochastic differential equations, condensed matter physics, statistical physics, chaotic systems, numerical linear algebra, neural networks, multivariate statistics, information theory, signal processing and small-world networks. This article provides a tutorial on random matrices which provides an overview of the theory and brings together in one source the most significant results recently obtained. Furthermore, the application of random matrix theory to the fundamental limits of wireless communication channels is described in depth.

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Random Matrix Theory and Wireless Communications

Real and Complex Analysis

The theory of random matrices plays an important role in many areas of pure mathematics and employs a variety of sophisticated mathematical tools (analytical, probabilistic and combinatorial). This diverse array of tools, while attesting to the vitality of the field, presents several formidable obstacles to the newcomer, and even the expert probabilist. This rigorous introduction to the basic theory is sufficiently self-contained to be accessible to graduate students in mathematics or related sciences, who have mastered probability theory at the graduate level, but have not necessarily been exposed to advanced notions of functional analysis, algebra or geometry. Useful background material is collected in the appendices and exercises are also included throughout to test the reader's understanding. Enumerative techniques, stochastic analysis, large deviations, concentration inequalities, disintegration and Lie algebras all are introduced in the text, which will enable readers to approach the research literature with confidence.

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An Introduction to Random Matrices

Part I. Basic Concepts: 1. Non-commutative probability spaces and distributions 2. A case study of non-normal distribution 3. C*-probability spaces 4. Non-commutative joint distributions 5. Definition and basic properties of free independence 6. Free product of *-probability spaces 7. Free product of C*-probability spaces Part II. Cumulants: 8. Motivation: free central limit theorem 9. Basic combinatorics I: non-crossing partitions 10. Basic Combinatorics II: Mobius inversion 11. Free cumulants: definition and basic properties 12. Sums of free random variables 13. More about limit theorems and infinitely divisible distributions 14. Products of free random variables 15. R-diagonal elements Part III. Transforms and Models: 16. The R-transform 17. The operation of boxed convolution 18. More on the 1-dimensional boxed convolution 19. The free commutator 20. R-cyclic matrices 21. The full Fock space model for the R-transform 22. Gaussian Random Matrices 23. Unitary Random Matrices Notes and Comments Bibliography Index.

Lectures on the Combinatorics of Free Probability

Preliminaries on non-crossing partitions Operator-valued multiplicative functions on the lattice of non-crossing partitions Amalgamated free products Operator-valued free probability theory Operator-valued stochastic processes and stochastic differential equations Bibliography.

Combinatorial Theory of the Free Product With Amalgamation and Operator-Valued Free Probability Theory

We present an example of a generalized Brownian motion. It is given by creation and annihilation operators on a “twisted” Fock space ofL2(ℝ). These operators fulfill (for a fixed −1≦μ≦1) the relationsc(f)c*(g)−μc*(g)c(f)=〈f,g〉1 (f, g ∈L2(ℝ)). We show that the distribution of these operators with respect to the vacuum expectation is a generalized Gaussian distribution, in the sense that all moments can be calculated from the second moments with the help of a combinatorial formula. We also indicate that our Brownian motion is one component of ann-dimensional Brownian motion which is invariant under the quantum groupS ν U(n) of Woronowicz (withμ =v2).

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An example of a generalized Brownian motion

The lattice of non-crossing partitions was introduced by Kreweras [Kre] in 1972 and examined further by Poupard [Pou] and Edelman [Edel, Ede2]. In the last time there has been an increased interest in this lattice, on one hand from the purely combinatorial point of view [SiU, EdS, Sire, BSS] and on the other hand from a (quantum) probabilistic point of view [GSS, Spel, Spe2, Spe3]. We became interested in this lattice when we noticed that it is connected with some new form of convolution for probability measures, the 'free' convolution introduced by Voiculescu [Voil, Voi2, Voi3]. This connection between the free convolution and the lattice of non-crossing partitions is exactly analogous to the connection between the usual convolution and the lattice of all partitions. We shall work out this connection in the last part of this work. The intention of this article is to show that both the purely combinatorial and the probabilistic point of view can benefit from each other. Motivated by Voiculescu's formula for the R-series of free convolution we examine multiplicative functions on the lattice of non-crossing partitions and prove our main theorem, which is much in the spirit of Voiculescu's formula. We shall then use this main theorem for deriving some known results on non-crossing partitions in a unified way. This part of our work is purely combinatorial, the free convolution serves only as a motivation for our theorem. In the last section we shall switch to the probabilistie side and show how the lattice of non-crossing partitions determines the structure of the free convolution. Voiculescu's formula will then follow as an easy corollary of our main theorem. In this respect, we shall get a purely combinatorial proof of this formula without any operator algebraic or functional analytic tools.

Multiplicative functions on the lattice of non-crossing partitions and free convolution.

We examine, for −1<q<1, q-Gaussian processes, i.e. families of operators (non-commutative random variables) – where the a

 t
 fulfill the q-commutation relations for some covariance function – equipped with the vacuum expectation state. We show that there is a q-analogue of the Gaussian functor of second quantization behind these processes and that this structure can be used to translate questions on q-Gaussian processes into corresponding (and much simpler) questions in the underlying Hilbert space. In particular, we use this idea to show that a large class of q-Gaussian processes possesses a non-commutative kind of Markov property, which ensures that there
exist classical versions of these non-commutative processes. This answers an old question of Frisch and Bourret [FB].

Roland Speicher

Papers

Lectures on the Combinatorics of Free Probability

Combinatorial Theory of the Free Product With Amalgamation and Operator-Valued Free Probability Theory

An example of a generalized Brownian motion

Multiplicative functions on the lattice of non-crossing partitions and free convolution.

q-Gaussian Processes: Non-commutative and Classical Aspects