We extend Natural Deduction for intuitionistic logic with a third introduction rule for the disjunction, $\vee$-i3, with a conclusion $Γ\vdash A\vee B$, but both premises $Γ\vdash A$ and $Γ\vdash B$. This rule is admissible in Natural Deduction. This extension is interesting in several respects. First, it permits to solve a well-known problem in logics with interstitial rules that have a weak introduction property: closed cut-free proofs end with an introduction rule, except in the case of disjunctions. With this new introduction rule, we recover the strong introduction property: closed cut-free proofs always end with an introduction. Second, the termination proof of this proof system is simpler than that of the usual propositional Natural Deduction with interstitial rules, as it does not require the use of the so-called ultra-reduction rules. Third, this proof system, in its linear version, has applications to quantum computing: the $\vee$-i3 rule enables the expression of quantum measurement, without the cost of introducing a new connective. Finally, even in logics without interstitial rules, the rule $\vee$-i3 is useful to reduce commuting cuts, although, in this paper, we leave
These are course notes for the 'Introduction to holography' Master level course at University of Cologne. The goal of the course is to give a pedogogical introduction to holography. Holography is a popular approach to quantum gravity, in which a theory of gravity can be described by a lower-dimensional boundary theory that itself has no gravity. The most concrete known example of a holographic model is the AdS/CFT correspondence, where the gravitational theory has a negative cosmological constant (the universe is asymptotically Anti-de Sitter) and the boundary theory is a conformal field theory. Symmetry plays a very important role in this duality. We therefore start the course with a review of Poincaré symmetry in quantum field theory, before moving on in the second chapter to conformal symmetry in conformally invariant quantum field theories or CFT's. Then we move to the basics of AdS physics in chapters 3 and 4, which will already reveal hints to the existence of a duality with CFT. After gathering the basic ingredients (CFT and AdS), in the second half of the course we are ready to formulate the AdS/CFT correspondence (chapter 5), including finite temperature AdS/CFT (chapter 6
These notes are an introduction to the theory of quantum symmetries of finite and infinite sets, graphs, and locally compact spaces.
This paper provides a preparatory introduction to torsors, written with a view toward later applications in the author's work. Rather than aiming at a comprehensive survey, the exposition focuses on those aspects of torsors that are most useful for understanding torsor-based reasoning: group actions, orbits, free transitive actions, the absence of a canonically chosen origin, and the interpretation of group elements as transports between points. After developing the basic definition and several elementary examples, we emphasize a central theme: torsors are not only characterized abstractly by free transitive group actions, but also arise naturally as objects obtained by gluing local trivial pieces by means of transition data satisfying cocycle conditions. A brief optional section indicates a sheaf- and topos-theoretic perspective. In the final part, we explain how these ideas prepare the ground for later conceptual applications, including aspects of $Σ$-protocols.
While Variational Inference (VI) is central to modern generative models like Variational Autoencoders (VAEs) and Denoising Diffusion Models (DDMs), its pedagogical treatment is split across disciplines. In statistics, VI is typically framed as a Bayesian method for posterior approximation. In machine learning, however, VAEs and DDMs are developed from a Frequentist viewpoint, where VI is used to approximate a maximum likelihood estimator. This creates a barrier for statisticians, as the principles behind VAEs and DDMs are hard to contextualize without a corresponding Frequentist introduction to VI. This paper provides that introduction: we explain the theory for VI, VAEs, and DDMs from a purely Frequentist perspective, starting with the classical Expectation-Maximization (EM) algorithm. We show how VI arises as a scalable solution for intractable E-steps and how VAEs and DDMs are natural, deep-learning-based extensions of this framework, thereby bridging the gap between classical statistical inference and modern generative AI.
This is an introduction to algebraic combinatorics, written for a quarter-long graduate course. It starts with a rigorous introduction to formal power series with some combinatorial applications, then discusses integer partitions (proving Jacobi's triple product identity), permutations (Lehmer codes, cycles) and subtractive methods (alternating sums, cancellations and inclusion-exclusion principles, with a particular focus on sign-reversing involutions and determinants). The last chapter introduces symmetric polynomials and proves the Littlewood--Richardson rule using Bender--Knuth involutions (a la Stembridge). The appendix contains over 200 exercises (without solutions).
The Quantum Approximate Optimization Algorithm (QAOA) is a promising variational quantum algorithm introduced to tackle classically intractable combinatorial optimization problems. This tutorial offers a comprehensive, first-principles introduction to QAOA and its properties, focusing on its application to Quadratic and Polynomial Unconstrained Binary Optimization (QUBO and PUBO) problems. The tutorial begins by outlining variational quantum circuits and QUBO problems, focusing on their key properties and the encoding of problem constraints through quadratic penalty terms. Next, it explores the QAOA in detail, covering its Hamiltonian formulation, gate decomposition, and example applications, along with their implementation and performance results. This is followed by an analysis of the algorithm's energy landscape, where proofs are provided for its symmetry and periodicity, and where a resulting parameter space reduction is proposed. Finally, the tutorial extends these concepts to PUBO problems by generalizing the results to higher-order Hamiltonians and discussing the associated symmetries and circuit construction.
These Lecture Notes are a brief introduction to the Malliavin calculus. In particular, different notions of Malliavin derivative found in the literature are considered and compared.
This paper provides an introduction to Kundt spaces, clarifying several important properties, many of which are typically scattered across the mathematical literature or presented without explicit reference to Kundt terminology. While not exhaustive, our approach aims to offer a pedagogical introduction, using a more geometric language and focusing on key concepts directly related to these spaces, such as lightlike totally geodesic foliations.
This book provides a self-contained introduction to geometric group theory. The topics range from an introduction of Cayley and Schreier graphs to Gromov's theorem on groups of polynomial growth and amenability. We discuss the ping-pong lemma, quasi-isometries, growth of groups, hyperbolicity, and other related notions. The book is based on graduate courses and can be used for such a course or for independent study.
This paper provides an introduction to quantum machine learning, exploring the potential benefits of using quantum computing principles and algorithms that may improve upon classical machine learning approaches. Quantum computing utilizes particles governed by quantum mechanics for computational purposes, leveraging properties like superposition and entanglement for information representation and manipulation. Quantum machine learning applies these principles to enhance classical machine learning models, potentially reducing network size and training time on quantum hardware. The paper covers basic quantum mechanics principles, including superposition, phase space, and entanglement, and introduces the concept of quantum gates that exploit these properties. It also reviews classical deep learning concepts, such as artificial neural networks, gradient descent, and backpropagation, before delving into trainable quantum circuits as neural networks. An example problem demonstrates the potential advantages of quantum neural networks, and the appendices provide detailed derivations. The paper aims to help researchers new to quantum mechanics and machine learning develop their expertise mo
While many good textbooks are available on Protein Structure, Molecular Simulations, Thermodynamics and Bioinformatics methods in general, there is no good introductory level book for the field of Structural Bioinformatics. This book aims to give an introduction into Structural Bioinformatics, which is where the previous topics meet to explore three dimensional protein structures through computational analysis. We provide an overview of existing computational techniques, to validate, simulate, predict and analyse protein structures. More importantly, it will aim to provide practical knowledge about how and when to use such techniques. We will consider proteins from three major vantage points: Protein structure quantification, Protein structure prediction, and Protein simulation & dynamics. Within the living cell, protein molecules perform specific functions, typically by interacting with other proteins, DNA, RNA or small molecules. They take on a specific three dimensional structure, encoded by its amino acid sequence, which allows them to function within the cell. Hence, the understanding of a protein's function is tightly coupled to its sequence and its three dimensional stru
This book provides an introduction to string field theory (SFT). String theory is usually formulated in the worldsheet formalism, which describes a single string (first-quantization). While this approach is intuitive and could be pushed far due to the exceptional properties of two-dimensional theories, it becomes cumbersome for some questions or even fails at a more fundamental level. These motivations have led to the development of SFT, a description of string theory using the field theory formalism (second-quantization). As a field theory, SFT provides a rigorous and constructive formulation of string theory. The main objective is to construct the closed bosonic SFT and to explain how to assess the consistency of string theory with it. The accent is put on providing the reader with the foundations, conceptual understanding and intuition of what SFT is. After reading this book, they should be able to study the applications from the literature. The book is organized in two parts. The first part reviews the topics of the worldsheet theory that are necessary to build SFT (worldsheet path integral, CFT and BRST quantization). The second part starts by introducing general concepts of S
We give a brief introduction to the software KnotPlot. The goals of this chapter are twofold: 1) to help a new user get started with using KnotPlot and 2) to provide veteran users with additional background and functionality available in the software.
While many good textbooks are available on Protein Structure, Molecular Simulations, Thermodynamics and Bioinformatics methods in general, there is no good introductory level book for the field of Structural Bioinformatics. This book aims to give an introduction into Structural Bioinformatics, which is where the previous topics meet to explore three dimensional protein structures through computational analysis. We provide an overview of existing computational techniques, to validate, simulate, predict and analyse protein structures. More importantly, it will aim to provide practical knowledge about how and when to use such techniques. We will consider proteins from three major vantage points: Protein structure quantification, Protein structure prediction, and Protein simulation & dynamics. In this chapter we explore basic physical and chemical concepts required to understand protein folding. We introduce major (de)stabilising factors of folded protein structures such as the hydrophobic effect and backbone entropy. In addition, we consider different states along the folding pathway, as well as natively disordered proteins and aggregated protein states. In this chapter, an intuit
This is a book about Lieb's Simplified approach to the Bose gas, which is a family of effective single-particle equations to study the ground state of many-body systems of interacting Bosons. It was introduced by Lieb in 1963, and recently found to have some rather intriguing properties. One of the equations of the approach, called the Simple equation, has been proved to make a prediction for the ground state energy that is asymptotically accurate both in the low- and the high-density regimes. Its predictions for the condensate fraction, two-point correlation function, and momentum distribution also agree with those of Bogolyubov theory at low density, despite the fact that it is based on ideas that are very different from those of Bogolyubov theory. In addition, another equation of the approach called the Big equation has been found to yield numerically accurate results for these observables over the entire range of densities for certain interaction potentials. This book is an introduction to Lieb's Simplified approach, and little background knowledge is assumed. We begin with a discussion of Bose gases and quantum statistical mechanics, and the notion of Bose-Einstein condensatio
The scope of this teaching package is to make a brief introduction to some notions and properties of chaotic systems. We first make a brief introduction to chaos in general and then we show some important properties of chaotic systems using the logistic map and its bifurcation diagram. We also show the universality found in "the route to chaos". The user is only required to have notions of algebra, so it is quite accessible. The formal basis of chaos theory are not covered in this introduction, but are pointed out for the reader interested in them. Therefore, this package is also useful for people who are interested in going deep into the mathematical theories, because it is a simple introduction of the terminology, and because it points out which are the original sources of information (so there is no danger in falling in the trap of "Learn Chaos in 48 hours" or "Bifurcation Diagrams for Dummies"). The included exercises are suggested for consolidating the covered topics. The on-line resources are highly recommended for extending this brief induction.
We provide an introduction to molecular dynamics simulations in the context of the Kob-Andersen model of a glass. We introduce a complete set of tools for doing and analyzing the results of simulations at fixed NVE and NVT. The modular format of the paper allows readers to select sections that meet their needs. We start with an introduction to molecular dynamics independent of the programming language, followed by introductions to an implementation using Python and then the freely available open source software package LAMMPS. We also describe analysis tools for the quick testing of the program during its development and compute the radial distribution function and the mean square displacement using both Python and LAMMPS.
These lecture notes provide an introduction to logarithmic geometry with a view towards recent applications in the desingularization theory.
This note provides a brief introduction to Redis highlighting its usefulness in multi-lingual statistical computing.