Pairwise Compatibility for 2-Simple Minded Collections

In $\tau$-tilting theory, it is often difficult to determine when a set of bricks forms a 2-simple minded collection. The aim of this paper is to determine when a set of bricks is contained in a 2-simple minded collection for a $\tau$-tilting finite algebra. We begin by extending the definition of mutation from 2-simple minded collections to more general sets of bricks (which we call semibrick pairs). This gives us an algorithm to check if a semibrick pair is contained in a 2-simple minded collection. We then use this algorithm to show that the 2-simple minded collections of a $\tau$-tilting finite gentle algebra (whose quiver contains no loops or 2-cycles) are given by pairwise compatibility conditions if and only if every vertex in the corresponding quiver has degree at most 2. As an application, we show that the classifying space of the $\tau$-cluster morphism category of a $\tau$-tilting finite gentle algebra (whose quiver contains no loops or 2-cycles) is an Eilenberg-MacLane space if every vertex in the corresponding quiver has degree at most 2.Comment: v4: changed title, updated and added references, changes to exposition for clarity and readability, corrected typos. v3: changes to introduction. v2: Made a correction to Corollary 5.12 (Theorem D in the introduction). 27 page

{\tau}-Cluster Morphism Categories and Picture Groups

$\tau$-cluster morphism categories were introduced by Buan and Marsh as a generalization of cluster morphism categories (as defined by Igusa and Todorov) to $\tau$-tilting finite algebras. In this paper, we show that the classifying space of such a category is a cube complex, generalizing results of Igusa and Todorov and Igusa. We further show that the fundamental group of this space is isomorphic to a generalized version of the picture group of the algebra, as defined by Igusa, Todorov, and Weyman. We end this paper by showing that if the algebra is Nakayama, then this space is locally $\mathsf{CAT}(0)$, and hence a $K(\pi,1)$. We do this by constructing a combinatorial interpretation of the 2-simple minded collections for Nakayama algebras. A key step in the proof is to show that, for Nakayama algebras, 2-simple minded collections are characterized by pairwise compatibility conditions, a fact not true in general.Comment: 29 pages, 14 figures. v3: Numerous improvements have been made following suggestions of an anonymous referee. v2: Lemma 4.13 has been combined with Lemma 4.12 (now Lemma 4.11) and its proof has been changed. The introduction has been rewritten and minor typos have been fixe

OMS FDIR: Initial prototyping

The Space Station Freedom Program (SSFP) Operations Management System (OMS) will automate major management functions which coordinate the operations of onboard systems, elements and payloads. The objectives of OMS are to improve safety, reliability and productivity while reducing maintenance and operations cost. This will be accomplished by using advanced automation techniques to automate much of the activity currently performed by the flight crew and ground personnel. OMS requirements have been organized into five task groups: (1) Planning, Execution and Replanning; (2) Data Gathering, Preprocessing and Storage; (3) Testing and Training; (4) Resource Management; and (5) Caution and Warning and Fault Management for onboard subsystems. The scope of this prototyping effort falls within the Fault Management requirements group. The prototyping will be performed in two phases. Phase 1 is the development of an onboard communications network fault detection, isolation, and reconfiguration (FDIR) system. Phase 2 will incorporate global FDIR for onboard systems. Research into the applicability of expert systems, object-oriented programming, fuzzy sets, neural networks and other advanced techniques will be conducted. The goals and technical approach for this new SSFP research project are discussed here

Landauer's Principle in Repeated Interaction Systems

We study Landauer's Principle for Repeated Interaction Systems (RIS) consisting of a reference quantum system $\mathcal{S}$ in contact with a structured environment $\mathcal{E}$ made of a chain of independent quantum probes; $\mathcal{S}$ interacts with each probe, for a fixed duration, in sequence. We first adapt Landauer's lower bound, which relates the energy variation of the environment $\mathcal{E}$ to a decrease of entropy of the system $\mathcal{S}$ during the evolution, to the peculiar discrete time dynamics of RIS. Then we consider RIS with a structured environment $\mathcal{E}$ displaying small variations of order $T^{-1}$ between the successive probes encountered by $\mathcal{S}$, after $n\simeq T$ interactions, in keeping with adiabatic scaling. We establish a discrete time non-unitary adiabatic theorem to approximate the reduced dynamics of $\mathcal{S}$ in this regime, in order to tackle the adiabatic limit of Landauer's bound. We find that saturation of Landauer's bound is equivalent to a detailed balance condition on the repeated interaction system, reflecting the non-equilibrium nature of the repeated interaction system dynamics. This is to be contrasted with the generic saturation of Landauer's bound known to hold for continuous time evolution of an open quantum system interacting with a single thermal reservoir in the adiabatic regime.Comment: Linked entropy production to detailed balance relation, improved presentation, and added concluding sectio

Human Sensing Expansion for a Rolling Robot

For the past twenty years, fans of the Star Wars movies have been building their own replica robots or “droids” that they fell in love with on the big screen. This project takes an existing design for the BB-8 droid and integrates sensors that can detect people and then interact with them. This design uses six IR sensors and one main 8x8 IR sensor grid to be able to detect the closest person and turn BB- 8’s dome towards them as though they are looking at them

Entropic Continuity Bounds & Eventually Entanglement-Breaking Channels

This thesis combines two parallel research directions: an exploration into the continuity properties of certain entropic quantities, and an investigation into a simple class of physical systems whose time evolution is given by the repeated application of a quantum channel. In the first part of the thesis, we present a general technique for establishing local and uniform continuity bounds for Schur concave functions; that is, for real-valued functions which are decreasing in the majorization pre-order. Continuity bounds provide a quantitative measure of robustness, addressing the following question: If there is some uncertainty or error in the input, how much uncertainty is there in the output? Our technique uses a particular relationship between majorization and the trace distance between quantum states (or total variation distance, in the case of probability distributions). Namely, the majorization pre-order attains a maximum and a minimum over ε-balls in this distance. By tracing the path of the majorization-minimizer as a function of the distance ε, we obtain the path of ``majorization flow’’. An analysis of the derivatives of Schur concave functions along this path immediately yields tight continuity bounds for such functions. In this way, we find a new proof of the Audenaert-Fannes continuity bound for the von Neumann entropy, and the necessary and sufficient conditions for its saturation, in a universal framework which extends to the other functions, including the Rényi and Tsallis entropies. In particular, we prove a novel uniform continuity bound for the α-Rényi entropy with α > 1 with much improved dependence on the dimension of the underlying system and the parameter α compared to previously known bounds. We show that this framework can also be used to provide continuity bounds for other Schur concave functions, such as the number of connected components of a certain random graph model as a function of the underlying probability distribution, and the number of distinct realizations of a random variable in some fixed number of independent trials as a function of the underlying probability mass function. The former has been used in modeling the spread of epidemics, while the latter has been studied in the context of estimating measures of biodiversity from observations; in these contexts, our continuity bounds provide quantitative estimates of robustness to noise or data collection errors. In the second part, we consider repeated interaction systems, in which a system of interest interacts with a sequence of probes, i.e. environmental systems, one at a time. The state of the system after each interaction is related to the state of the system before the interaction by the so-called reduced dynamics, which is described by the action of a quantum channel. When each probe and the way it interacts with the system is identical, the reduced dynamics at each step is identical. In this scenario, under the additional assumption that the reduced dynamics satisfies a faithfulness property, we characterize which repeated interaction systems break any initially-present entanglement between the system and an untouched reference, after finitely many steps. In this case, the reduced dynamics is said to be eventually entanglement-breaking. This investigation helps improve our understanding of which kinds of noisy time evolution destroy entanglement. When the probes and their interactions with the system are slowly-varying (i.e. adiabatic), we analyze the saturation of Landauer's bound, an inequality between the entropy change of the system and the energy change of the probes, in the limit in which the number of steps tends to infinity and both the difference between consecutive probes and the difference between their interactions vanishes. This analysis proceeds at a fine-grained level by means of a two-time measurement protocol, in which the energy of the probes is measured before and after each interaction. The quantities of interest are then studied as random variables on the space of outcomes of the energy measurements of the probes, providing a deeper insight into the interrelations between energy and entropy in this setting.Cantab Capital Institute for the Mathematics of Informatio

On Sign-Solvable Linear Systems and their Applications in Economics

Sign-solvable linear systems are part of a branch of mathematics called qualitative matrix theory. Qualitative matrix theory is a development of matrix theory based on the sign (¡; 0; +) of the entries of a matrix. Sign-solvable linear systems are useful in analyzing situations in which quantitative data is unknown or had to measure, but qualitative information is known. These situations arise frequently in a variety of disciplines outside of mathematics, including economics and biology. The applications of sign-solvable linear systems in economics are documented and the development of new examples is formalized mathematically. Additionally, recent mathematical developments about sign-solvable linear systems and their implications to economics are discussed

State level trends in renewable energy procurement via solar installation versus green electricity

“In the past 5 years, consumer options for procuring renewable energy have increased, ranging from rooftop solar installation to utility green pricing to Community Choice Aggregation. These options vary in terms of costs and benefits to the consumer as well as grid integration implications. However, little is known regarding how the presence of a wide range of options for utility-scale renewable procurement affects demand for distributed residential solar installations. In theory, there are three possible relationships, (1) positive correlation, where utility-scale and distributed resources complement each other to increase overall production, (2) negative correlation, where utility-scale and distributed resources are substitutes, and (3) no correlation, suggesting that these different procurement choices are unrelated. To examine the relationship, aggregated at the state level, we use a mixed effects regression model with panel data from 2016 to 2019 for all fifty US states plus the District of Columbia, controlling for policy, resource availability, and demographics. Although there was no evidence of a relationship between demand for utility-scale and distributed options across states, the estimated random effects suggest variation between states. An investigation of Vermont (positive), North Dakota (negative), and Oregon (zero) suggest that the policy environment, available resources, and average energy cost may explain this heterogeneity. As more data becomes available over time, there will be additional opportunities to explore this relationship”--Abstract, page iii