Multi-point distribution of discrete time periodic TASEP

We study the one-dimensional discrete time totally asymmetric simple exclusion process with parallel update rules on a spatially periodic domain. A multi-point space-time joint distribution formula is obtained for general initial conditions. The formula involves contour integrals of Fredholm determinants with kernels acting on certain discrete spaces. For a class of initial conditions satisfying certain technical assumptions, we are able to derive large-time, large-period limit of the joint distribution, under the relaxation time scale $$t=O(L^{3/2})$$ t = O ( L 3 / 2 ) when the height fluctuations are critically affected by the finite geometry. The assumptions are verified for the step and flat initial conditions. As a corollary we obtain the multi-point distribution of discrete time TASEP on the whole integer lattice $${\mathbb {Z}}$$ Z by taking the period L large enough so that the finite-time distribution is not affected by the boundary. The large time limit for the multi-time distribution of discrete time TASEP on $${\mathbb {Z}}$$ Z is then obtained for the step initial condition.

Strongly Regular Graphs

Strongly regular graphs lie at the intersection of statistical design, group theory, finite geometry, information and coding theory, and extremal combinatorics. This monograph collects all the major known results together for the first time in book form, creating an invaluable text that researchers in algebraic combinatorics and related areas will refer to for years to come. The book covers the theory of strongly regular graphs, polar graphs, rank 3 graphs associated to buildings and Fischer groups, cyclotomic graphs, two-weight codes and graphs related to combinatorial configurations such as Latin squares, quasi-symmetric designs and spherical designs. It gives the complete classification of rank 3 graphs, including some new constructions. More than 100 graphs are treated individually. Some unified and streamlined proofs are featured, along with original material including a new approach to the (affine) half spin graphs of rank 5 hyperbolic polar spaces.

Some Proposed Problems on Permutation Polynomials over Finite Fields

From the 19th century, the theory of permutation polynomial over finite fields, that are arose in the work of Hermite and Dickson, has drawn general attention. Permutation polynomials over finite fields are an active area of research due to their rising applications in mathematics and engineering. The last three decades has seen rapid progress on the research on permutation polynomials due to their diverse applications in cryptography, coding theory, finite geometry, combinatorics and many more areas of mathematics and engineering. For this reason, the study of permutation polynomials is important nowadays. In this chapter, we propose some new problems in connection to permutation polynomials over finite fields by the help of prime numbers.

Generalized threshold secret sharing and finite geometry

In the history of secret sharing schemes many constructions are based on geometric objects. In this paper we investigate generalizations of threshold schemes and related finite geometric structures. In particular, we analyse compartmented and hierarchical schemes, and deduce some more general results, especially bounds for special arcs and novel constructions for conjunctive 2-level and 3-level hierarchical schemes.

ON THE PRODUCT OF ELEMENTS WITH PRESCRIBED TRACE

This paper deals with the following problem. Given a finite extension of fields $\mathbb{L}/\mathbb{K}$ and denoting the trace map from $\mathbb{L}$ to $\mathbb{K}$ by $\text{Tr}$ , for which elements $z$ in $\mathbb{L}$ , and $a$ , $b$ in $\mathbb{K}$ , is it possible to write $z$ as a product $xy$ , where $x,y\in \mathbb{L}$ with $\text{Tr}(x)=a,\text{Tr}(y)=b$ ? We solve most of these problems for finite fields, with a complete solution when the degree of the extension is at least 5. We also have results for arbitrary fields and extensions of degrees 2, 3 or 4. We then apply our results to the study of perfect nonlinear functions, semifields, irreducible polynomials with prescribed coefficients, and a problem from finite geometry concerning the existence of certain disjoint linear sets.

Towards the signs of new physics through the spectral action

We review recent results obtained by using the spectral action applied to some simple physical models built in the noncommutative geometry framework. The first result, based on the generalization of the spectral action principle to the case of the fermionic action leads to the lowest order corrections that can possibly explain the huge difference between the electron and neutrino masses. In the second case, the extension of the family of Dirac operators for the product of finite geometry with the Riemannian manifold but without a product metric leads to a theory with two metrics that is similar to the models of bimetric gravity.

Truncated Hilbert Space Approach

Chapter 25 covers the Truncated Hilbert Space Approach provides a very efficient numerical algorithm to study many properties of a perturbed conformal field theory defined on a finite geometry, typically an infinite cylinder of radius R. These include the masses of the various excitations, their number below threshold, the presence of false vacua and resonances, on-shell three-particle coupling, etc. The implementation of this approach does not depend on the integrability of the off-critical model and therefore it is a very useful tool to extend study to the entire scaling region around the critical point. This chapter discusses the basis of such an algorithm and provides some interesting applications thereof.

The Barreme Basin and the Gevaudan diapir - an example of the interplay between compressional tectonics and salt diapirism

<p><strong>The Barreme Basin and the Gevaudan diapir - an example of the interplay between compressional tectonics and salt diapirism </strong></p><p><strong> </strong></p><p><strong>Adam Csicsek and Rod Graham</strong></p><p>Imperial College London</p><p><strong> </strong></p><p>Our understanding of the role of salt diapirism in determining the finite geometry of fold and thrust belts has grown apace in the last few years, but the interplay between the two remains a significant problem for structural interpretation. The Gevaudan diapir in the fold and thrust belt of the sub-Alpine chain of Haute Provence is well known and has been documented by numerous eminent alpine structural geologists. Graciansky, Dardot, Mascle, Gidon and Lickorish and Ford have all described and illustrated the geometry and evolution of the structure, and Lickorish and Ford’s interpretation is figured as an example of  diapirism  in a compressional setting by Jackson and Hudec in their text on salt tectonics. We review these various interpretations and present another.</p><p>The differences between the various interpretations say much about the complex interplay of salt diapirism and thin-skinned thrusting and have profound implications for the way we interpret the tectonic and sedimentary evolution of the Barreme basin which lies adjacent to the diapir</p><p>The Barreme basin is a thrust-top fragment of the Provencal foreland basin and has been described in detail from both sedimentological (e.g. Evans and Elliott, 1999) and structural (e.g. Antoni and Meckel, 1997) points of view. Here we make the case that it is also a salt related minibasin - a secondary minibasin developed on a now welded allochthonous Middle Cretaceous salt canopy.  We believe that within the basin it is possible to interpret successive depocentres which may record progressive salt withdrawal. We argue that though thrust loading must be the fundamental driving mechanism responsible for salt movement late in the tectonic history of the region, thrusting has not done much more than modify existing salt related geometry.    </p>