Compact forms of reduced density matrices

10 págs.; 3 tabs.; PACS number~s!: 31.15.Hz, 31.10.1zThe necessary and sufficient minimum information carried by reduced density matrices (RDM) are discussed. A method is reported for obtaining the same information as a p-RDM although in compact form, from which all the redundant information is omitted. The algebra operations and basic properties of these compact-form matrices are obtained. ©2003 The American Physical SocietyWe are greatly indebted to the Spanish Ministerio de Educación, Cultura y Deporte for its support under Project No. BQU2000-1158Peer Reviewe

Modelling biomolecules through atomistic graphs: theory, implementation, and applications

Describing biological molecules through computational models enjoys ever-growing popularity. Never before has access to computational resources been easier for scientists across the natural sciences. The need for accurate, efficient, and robust modelling tools is therefore irrefutable. This, in turn, calls for highly interdisciplinary research, which the thesis presented here is a product of. Through the successful marriage of techniques from mathematical graph theory, theoretical insights from chemistry and biology, and the tools of modern computer science, we are able to computationally construct accurate depictions of biomolecules as atomistic graphs, in which individual atoms become nodes and chemical bonds/interactions are represented by weighted edges. When combined with methods from graph theory and network science, this approach has previously been shown to successfully reveal various properties of proteins, such as dynamics, rigidity, multi-scale organisation, allostery, and protein-protein interactions, and is well poised to set new standards in terms of computational feasibility, multi-scale resolution (from atoms to domains) and time-scales (from nanoseconds to milliseconds). Therefore, building on previous work in our research group spanning over 15 years and to further encourage and facilitate research into this growing field, this thesis's main contribution is to provide a formalised foundation for the construction of atomistic graphs. The most crucial aspect of constructing atomistic graphs of large biomolecules compared to small molecules is the necessity to include a variety of different types of bonds and interactions, because larger biomolecules attain their unique structural layout mainly through weaker interactions, e.g. hydrogen bonds, the hydrophobic effect or π-π interactions. Whilst most interaction types are well-studied and have readily available methodology which can be used to construct atomistic graphs, this is not the case for hydrophobic interactions. To fill this gap, the work presented herein includes novel methodology for encoding the hydrophobic effect in atomistic graphs, that accounts for the many-body effect and non-additivity. Then, a standalone software package for constructing atomistic graphs from structural data is presented. Herein lies the heart of this thesis: the combination of a variety of methodologies for a range of bond/interaction types, as well as an implementation that is deterministic, easy-to-use and efficient. Finally, some promising avenues for utilising atomistic graphs in combination with graph theoretical tools such as Markov Stability as well as other approaches such as Multilayer Networks to study various properties of biomolecules are presented.Open Acces

Modern Machine Learning for LHC Physicists

Modern machine learning is transforming particle physics, faster than we can follow, and bullying its way into our numerical tool box. For young researchers it is crucial to stay on top of this development, which means applying cutting-edge methods and tools to the full range of LHC physics problems. These lecture notes are meant to lead students with basic knowledge of particle physics and significant enthusiasm for machine learning to relevant applications as fast as possible. They start with an LHC-specific motivation and a non-standard introduction to neural networks and then cover classification, unsupervised classification, generative networks, and inverse problems. Two themes defining much of the discussion are well-defined loss functions reflecting the problem at hand and uncertainty-aware networks. As part of the applications, the notes include some aspects of theoretical LHC physics. All examples are chosen from particle physics publications of the last few years. Given that these notes will be outdated already at the time of submission, the week of ML4Jets 2022, they will be updated frequently.Comment: First version, we very much appreciate feedbac

Polyprograms and Polyprogram Bisimulation

A polyprogram is a generalization of a program which admits multiple definitions of a single function. Such objects arise in different transformation systems, such as the Burstall-Darlington framework or equality saturation. In this paper, we introduce the notion of a polyprogram in a non-strict first-order functional language. We define denotational semantics for polyprograms and describe some possible transformations of polyprograms, namely we present several main transformations in two different styles: in the style of the Burstall-Darlington framework and in the style of equality saturation. Transformations in the style of equality saturation are performed on polyprograms in decomposed form, where the difference between functions and expressions is blurred, and so is the difference between substitution and unfolding. Decomposed polyprograms are well suited for implementation and reasoning, although they are not very human-readable. We also introduce the notion of polyprogram bisimulation which enables a powerful transformation called merging by bisimulation, corresponding to proving equivalence of functions by induction or coinduction. Polyprogram bisimulation is a concept inspired by bisimulation of labelled transition systems, but yet it is quite different, because polyprogram bisimulation treats every definition as self-sufficient, that is a function is considered to be defined by any of its definitions, whereas in an LTS the behaviour of a state is defined by all transitions from this state. We present an algorithm for enumerating polyprogram bisimulations of a certain form. The algorithm consists of two phases: enumerating prebisimulations and converting them to proper bisimulations. This separation is required because polyprogram bisimulations take into account the possibility of parameter permutation. We prove correctness of this algorithm and formulate a certain weak form of its completeness. The article is published in the author’s wording

Refactoring proofs

Refactoring is an important Software Engineering technique for improving the structure of a program after it has been written. Refactorings improve the maintainability, readability, and design of a program without affecting its external behaviour. In analogy, this thesis introduces proof refactoring to make structured, semantics preserving changes to the proof documents constructed by interactive theorem provers as part of a formal proof development. In order to formally study proof refactoring, the first part of this thesis constructs a proof language framework, Hiscript. The Hiscript framework consists of a procedural tactic language, a declarative proof language, and a modular theory language. Each level of this framework is equipped with a formal semantics based on a hierarchical notion of proof trees. Furthermore, this framework is generic as it does not prescribe an underlying logical kernel. This part contributes an investigation of semantics for formal proof documents, which is proved to construct valid proofs. Moreover, in analogy with type-checking, static well-formedness checks of proof documents are separated from evaluation of the proof. Furthermore, a subset of the SSReflect language for Coq, called eSSence, is also encoded using hierarchical proofs. Both Hiscript and eSSence are shown to have language elements with a natural hierarchical representation. In the second part, proof refactoring is put on a formal footing with a definition using the Hiscript framework. Over thirty refactorings are formally specified and proved to preserve the semantics in a precise way for the Hiscript language, including traditional structural refactorings, such as rename item, and proof specific refactorings such as backwards proof to forwards proof and declarative to procedural. Finally, a concrete, generic refactoring framework, called Polar, is introduced. Polar is based on graph rewriting and has been implemented with over ten refactorings and for two proof languages, including Hiscript. Finally, the third part concludes with some wishes for the future

Tracking Extended Objects in Noisy Point Clouds with Application in Telepresence Systems

We discuss theory and application of extended object tracking. This task is challenging as sensor noise prevents a correct association of the measurements to their sources on the object, the shape itself might be unknown a priori, and due to occlusion effects, only parts of the object are visible at a given time. We propose an approach to track the parameters of arbitrary objects, which provides new solutions to the above challenges, and marks a significant advance to the state of the art

Modifying T cell differentiation via protein based delivery of signalling attenuators

In recent years, evidence has accumulated that hierarchical thresholds for proliferation and differentiation may determine the generation of various effector T cell modes. It is as yet poorly understood, however, how changes in TCR-mediated signalling and its downstream signalling cascades drive the differentiation of the naive T cell into different response modes. This project aims to study whether manipulation of the signalling pathways directly downstream of the TCR/CD28 receptor can result in distinct functional differentiation of monoclonal CD4+ T cell populations with known antigenic specificity. To this end, two potential protein transfer mechanisms were tested for their suitability to act as inert delivery vehicles to deliver proteins into ex vivo naive T cells. The B subunit pentamer of enterotoxin B induced high levels apoptosis in naive T cells, which precluded its use as an inert delivery vehicle. The protein transduction domain of the HIV-1 protein Tat, however, did not influence T cell physiology and was subsequently successfully tested for its ability to transduce ex vivo T cells genetically fused to large proteins. Two fusion partners, the kinase truncated forms of ZAP-70 and Lck, were tested for their ability to interfere with TCR-mediated signalling. The kinase- truncated form of ZAP-70 showed a concentration dependent effect on calcium mobilisation, which was reliant on the strength of CD3 crosslinking. The kinase- truncated form of Lck also altered calcium mobilisation in a concentration dependent manner, but independent of TCR strength. In addition, it was shown to influence T cell activation, to be capable of altering the signal transduction cascade after T cell priming, and to dramatically change the cytokine profile