2,988 research outputs found
Pairwise Compatibility for 2-Simple Minded Collections
In -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 -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 -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 -cluster morphism category of a -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
-cluster morphism categories were introduced by Buan and Marsh as a
generalization of cluster morphism categories (as defined by Igusa and Todorov)
to -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 , and hence a
. 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 in contact with a
structured environment made of a chain of independent quantum
probes; 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 to a decrease of entropy of the
system during the evolution, to the peculiar discrete time
dynamics of RIS. Then we consider RIS with a structured environment
displaying small variations of order between the
successive probes encountered by , after interactions,
in keeping with adiabatic scaling. We establish a discrete time non-unitary
adiabatic theorem to approximate the reduced dynamics of 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
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