Abstract:
A novel method of transplanting algebras of observables from de Sitter space to a large class of Robertson-Walker space-times is exhibited. It allows one to establish the existence of an abundance of local nets on these spaces which comply with a recently proposed condition of geometric modular action. The corresponding modular symmetry groups appearing in these examples also satisfy a condition of modular stability, which has been suggested as a substitute for the requirement of positivity of the energy in Minkowski space. Moreover, they exemplify the conjecture that the modular symmetry groups are generically larger than the isometry and conformal groups of the underlying space-times.

Abstract:
In this paper, research state space structure properities of normal chain from Ray-Knight theory on a given set E : If S_infty(omega) whose probability is 1 are empty set,then,the normal chain only has jumping type track, at the time, the transfer function must be the least and only determined by desity matrix; If S_infty(omega) whose probability is positive are not empty set, then,the track of normal chain is not jumping type track, and its transfer function cannot determine by density matrix.

Abstract:
First we prove existence of a fixed point for mappings defined on a complete modular space satisfying a general contractive inequality of integral type. Then we generalize fixed-point theorem for a quasicontraction mapping given by Khamsi (2008) and Ciric (1974). 1. Introduction In [1], Branciari established that a function defined on a complete metric space satisfying a contraction condition of the form has a unique attractive fixed point where is a Lebesgue-integrable mapping and . In [2], Rhoades extended this result to a quasicontraction function . The purpose of this paper is to extend these theorems in modular space. First, we introduce the notion of modular space. Definition 1.1. Let be an arbitrary vector space over or . A functional is called modular if(1) if and only if ; (2) for with , for all ;(3) if , , for all . If (2.14) in Definition 1.1 is replaced by for , with an , then the modular is called an -convex modular; and if , is called a convex modular. Definition 1.2. A modular defines a corresponding modular space, that is, the space is given by Definition 1.3. Let be a modular space. (1)A sequence in is said to be (a) -convergent to if as , (b) -Cauchy if as . (2) is -complete if any -Cauchy sequence is -convergent. (3)A subset is said to be -closed if for any sequence with then . denotes the closure of in the sense of . (4)A subset is called -bounded if where is called the -diameter of . (5)We say that has Fatou property if whenever (6) is said to satisfy the -condition if: as whenever as . Remark 1.4. Note that since does not satisfy a priori the triangle inequality, we cannot expect that if and are -convergent, respectively, to and then is -convergent to , neither that a -convergent sequence is -Cauchy. 2. Main Result Theorem 2.1. Let be a complete modular space, where satisfies the -condition. Assume that is an increasing and upper semicontinuous function satisfying Let be a nonnegative Lebesgue-integrable mapping which is summable on each compact subset of and such that for , and let be a mapping such that there are where , for each . Then has a unique fixed point in . Proof. First, we show that for , the sequence converges to 0. For , we have Consequently, is decreasing and bounded from below. Therefore converges to a nonnegative point . Now, if , then which is a contradiction, so and This concludes . Suppose that then there exist a and a sequence such that then we get the following contradiction: Now, we prove for each the sequence is a -Cauchy sequence. Assume that there is an such that for each there exist that , Then we choose

Abstract:
We interpolate the Gauss-Manin connection in p-adic families of nearly overconvergent modular forms. This gives a family of Maass-Shimura type differential operators from the space of nearly overconvergent modular forms of type r to the space of nearly overconvergent modular forms of type r + 1 with p-adic weight shifted by 2. Our construction is purely geometric, using Andreatta-Iovita-Stevens and Pilloni's geometric construction of eigencurves, and should thus generalize to higher rank groups.

Abstract:
In a series of papers we have been studying the geometric theta correspondence for non-compact arithmetic quotients of symmetric spaces associated to orthogonal groups. It is our overall goal to develop a general theory of geometric theta liftings in the context of the real differential geometry/topology of non-compact locally symmetric spaces of orthogonal and unitary groups which generalizes the theory of Kudla-Millson in the compact case. In this paper we study in detail the geometric theta lift for Hilbert modular surfaces. In particular, we will give a new proof and an extension (to all finite index subgroups of the Hilbert modular group) of the celebrated theorem of Hirzebruch and Zagier that the generating function for the intersection numbers of the Hirzebruch-Zagier cycles is a classical modular form of weight 2. In our approach we replace Hirzebuch's smooth complex analytic compactification $\tilde{X}$ of the Hilbert modular surface $X$ with the (real) Borel-Serre compactification $\bar{X}$. The various algebro-geometric quantities are then replaced by topological quantities associated to 4-manifolds with boundary. In particular, the "boundary contribution" in Hirzebruch-Zagier is replaced by sums of linking numbers of circles (the boundaries of the cycles) in the 3-manifolds of type Sol (torus bundle over a circle) which comprise the Borel-Serre boundary.

Abstract:
A condition of geometric modular action is proposed as a selection principle for physically interesting states on general space-times. This condition is naturally associated with transformation groups of partially ordered sets and provides these groups with projective representations. Under suitable additional conditions, these groups induce groups of point transformations on these space-times, which may be interpreted as symmetry groups. The consequences of this condition are studied in detail in application to two concrete space-times -- four-dimensional Minkowski and three-dimensional de Sitter spaces -- for which it is shown how this condition characterizes the states invariant under the respective isometry group. An intriguing new algebraic characterization of vacuum states is given. In addition, the logical relations between the condition proposed in this paper and the condition of modular covariance, widely used in the literature, are completely illuminated.

Abstract:
We study spontaneous symmetry breaking for field algebras on Minkowski space in the presence of a condition of geometric modular action (CGMA) proposed earlier as a selection criterion for vacuum states on general space-times. We show that any internal symmetry group must commute with the representation of the Poincare group (whose existence is assured by the CGMA) and each translation-invariant vector is also Poincare invariant. The subspace of these vectors can be centrally decomposed into pure invariant states and the CGMA holds in the resulting sectors. As positivity of the energy is not assumed, similar results may be expected to hold for other space--times.

Abstract:
Many entities managed by HEP Software Frameworks represent spatial (3-dimensional) real objects. Effective definition, manipulation and visualization of such objects is an indispensable functionality. GraXML is a modular Geometric Modeling toolkit capable of processing geometric data of various kinds (detector geometry, event geometry) from different sources and delivering them in ways suitable for further use. Geometric data are first modeled in one of the Generic Models. Those Models are then used to populate powerful Geometric Model based on the Java3D technology. While Java3D has been originally created just to provide visualization of 3D objects, its light weight and high functionality allow an effective reuse as a general geometric component. This is possible also thanks to a large overlap between graphical and general geometric functionality and modular design of Java3D itself. Its graphical functionalities also allow a natural visualization of all manipulated elements. All these techniques have been developed primarily (or only) for the Java environment. It is, however, possible to interface them transparently to Frameworks built in other languages, like for example C++. The GraXML toolkit has been tested with data from several sources, as for example ATLAS and ALICE detector description and ATLAS event data. Prototypes for other sources, like Geometry Description Markup Language (GDML) exist too and interface to any other source is easy to add.

Abstract:
The subject of this thesis is the modular group of automorphisms acting on the massive algebra of local observables having their support in bounded open subsets of Minkowski space. After a compact introduction to micro-local analysis and the theory of one-parameter groups of automorphisms, which are used exensively throughout the investigation, we are concerned with modular theory and its consequences in mathematics, e.g., Connes' cocycle theorem and classification of type III factors and Jones' index theory, as well as in physics, e.g., the determination of local von Neumann algebras to be hyperfinite factors of type III_1, the formulation of thermodynamic equilibrium states for infinite-dimensional quantum systems (KMS states) and the discovery of modular action as geometric transformations. However, our main focus are its applications in physics, in particular the modular action as Lorentz boosts on the Rindler wedge, as dilations on the forward light cone and as conformal mappings on the double cone. Subsequently, their most important implications in local quantum physics are discussed. The purpose of this thesis is to shed more light on the transition from the known massless modular action to the wanted massive one in the case of double cones. First of all the infinitesimal generator of the massive modular group is investigated, especially some assumptions on its structure are verified explicitly for the first time for two concrete examples. Then, two strategies for the calculation of group itself are discussed. Some formalisms and results from operator theory and the method of second quantisation used in this thesis are made available in the appendix.

Abstract:
In this paper we show that states, transitions and behavior of concurrent systems can often be modeled as sheaves over a suitable topological space. In this context, geometric logic can be used to describe which local properties (i.e. properties of individual systems) are preserved, at a global level, when interconnecting the systems. The main area of application is to modular verification of complex systems. We illustrate the ideas by means of an example involving a family of interacting controllers for trains on a rail track.