78 research outputs found
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Characterisation of the Set of Ground States of Uniformly Chaotic Finite-Range Lattice Models
Chaotic dependence on temperature refers to the phenomenon of divergence of
Gibbs measures as the temperature approaches a certain value. Models with
chaotic behaviour near zero temperature have multiple ground states, none of
which are stable. We study the class of uniformly chaotic models, that is,
those in which, as the temperature goes to zero, every choice of Gibbs measures
accumulates on the entire set of ground states. We characterise the possible
sets of ground states of uniformly chaotic finite-range models up to computable
homeomorphisms.
Namely, we show that the set of ground states of every model with
finite-range and rational-valued interactions is topologically closed and
connected, and belongs to the class of the arithmetical hierarchy.
Conversely, every -computable, topologically closed and connected set of
probability measures can be encoded (via a computable homeomorphism) as the set
of ground states of a uniformly chaotic two-dimensional model with finite-range
rational-valued interactions.Comment: 46 pages, 12 figure
Computability and Tiling Problems
In this thesis we will present and discuss various results pertaining to
tiling problems and mathematical logic, specifically computability theory. We
focus on Wang prototiles, as defined in [32]. We begin by studying Domino
Problems, and do not restrict ourselves to the usual problems concerning finite
sets of prototiles. We first consider two domino problems: whether a given set
of prototiles has total planar tilings, which we denote , or whether
it has infinite connected but not necessarily total tilings, (short for
`weakly tile'). We show that both , and
thereby both and are -complete. We also show that
the opposite problems, and (short for `Strongly Not Tile')
are such that and so both
and are both -complete. Next we give some consideration to the
problem of whether a given (infinite) set of prototiles is periodic or
aperiodic. We study the sets of periodic tilings, and of
aperiodic tilings. We then show that both of these sets are complete for the
class of problems of the form . We also present
results for finite versions of these tiling problems. We then move on to
consider the Weihrauch reducibility for a general total tiling principle
as well as weaker principles of tiling, and show that there exist Weihrauch
equivalences to closed choice on Baire space, . We also show
that all Domino Problems that tile some infinite connected region are Weihrauch
reducible to . Finally, we give a prototile set of 15
prototiles that can encode any Elementary Cellular Automaton (ECA). We make use
of an unusual tile set, based on hexagons and lozenges that we have not see in
the literature before, in order to achieve this.Comment: PhD thesis. 179 pages, 13 figure
Computable classifications of continuous, transducer, and regular functions
We develop a systematic algorithmic framework that unites global and local
classification problems for functional separable spaces and apply it to attack
classification problems concerning the Banach space C[0,1] of real-valued
continuous functions on the unit interval. We prove that the classification
problem for continuous (binary) regular functions among almost everywhere
linear, pointwise linear-time Lipshitz functions is -complete. We
show that a function is (binary)
transducer if and only if it is continuous regular; interestingly, this
peculiar and nontrivial fact was overlooked by experts in automata theory. As
one of many consequences, our -completeness result covers the class
of transducer functions as well. Finally, we show that the Banach space
of real-valued continuous functions admits an arithmetical
classification among separable Banach spaces. Our proofs combine methods of
abstract computability theory, automata theory, and functional analysis.Comment: Revised argument in Section 5; results unchange
Recommended from our members
Accelerating Materials Discovery with Machine Learning
As we enter the data age, ever-increasing amounts of human knowledge are being recorded in machine-readable formats.
This has opened up new opportunities to leverage data to accelerate scientific discovery.
This thesis focuses on how we can use historical and computational data to aid the discovery and development of new materials.
We begin by looking at a traditional materials informatics task -- elucidating the structure-function relationships of high-temperature cuprate superconductors.
One of the most significant challenges for materials informatics is the limited availability of relevant data.
We propose a simple calibration-based approach to estimate the apical and in-plane copper-oxygen distances from more readily available lattice parameter data to address this challenge for cuprate superconductors.
Our investigation uncovers a large, unexplored region of materials space that may yield cuprates with higher critical temperatures.
We propose two experimental avenues that may enable this region to be accessed.
Computational materials exploration is bottle-necked by our ability to provide input structures to feed our workflows.
Whilst \textit{ab-intio} structure identification is possible, it is computationally burdensome and we lack design rules for deciding where to target searches in high-throughput setups.
To address this, there is a need to develop tools that suggest promising candidates, enabling automated deployment and increased efficiency.
Machine learning models are well suited to this task, however, current approaches typically use hand-engineered inputs.
This means that their performance is circumscribed by the intuitions reflected in the chosen inputs.
We propose a novel way to formulate the machine learning task as a set regression problem over the elements in a material.
We show that our approach leads to higher sample efficiency than other well-established composition-based approaches.
Having demonstrated the ability of machine learning to aid in the selection of promising compound compositions, we next explore how useful machine learning might be for identifying fabrication routes.
Using a recently released data-mined data set of solid-state synthesis reactions, we design a two-stage model to predict the products of inorganic reactions.
We critically explore the performance of this model, showing that whilst the predictions fall short of the accuracy required to be chemically discriminative, the model provides valuable insights into understanding inorganic reactions.
Through careful investigation of the model's failure modes, we explore the challenges that remain in the construction of forward inorganic reaction prediction models and suggest some pathways to tackle the identified issues.
One of the principal ways that material scientists understand and categorise materials is in terms of their symmetries.
Crystal structure prototypes are assigned based on the presence of symmetrically equivalent sites known as Wyckoff positions.
We show that a powerful coarse-grained representation of materials structures can be constructed from the Wyckoff positions by discarding information about their coordinates within crystal structures.
One of the strengths of this representation is that it maintains the ability of structure-based methods to distinguish polymorphs whilst also allowing combinatorial enumeration akin to composition-based approaches.
We construct an end-to-end differentiable model that takes our proposed Wyckoff representation as input.
The performance of this approach is examined on a suite of materials discovery experiments showing that it leads to strong levels of enrichment in materials discovery tasks.
The research presented in this thesis highlights the promise of applying data-driven workflows and machine learning in materials discovery and development.
This thesis concludes by speculating about promising research directions for applying machine learning within materials discovery
Collected Papers (on Neutrosophic Theory and Applications), Volume VI
This sixth volume of Collected Papers includes 74 papers comprising 974 pages on (theoretic and applied) neutrosophics, written between 2015-2021 by the author alone or in collaboration with the following 121 co-authors from 19 countries: Mohamed Abdel-Basset, Abdel Nasser H. Zaied, Abduallah Gamal, Amir Abdullah, Firoz Ahmad, Nadeem Ahmad, Ahmad Yusuf Adhami, Ahmed Aboelfetouh, Ahmed Mostafa Khalil, Shariful Alam, W. Alharbi, Ali Hassan, Mumtaz Ali, Amira S. Ashour, Asmaa Atef, Assia Bakali, Ayoub Bahnasse, A. A. Azzam, Willem K.M. Brauers, Bui Cong Cuong, Fausto Cavallaro, Ahmet Çevik, Robby I. Chandra, Kalaivani Chandran, Victor Chang, Chang Su Kim, Jyotir Moy Chatterjee, Victor Christianto, Chunxin Bo, Mihaela Colhon, Shyamal Dalapati, Arindam Dey, Dunqian Cao, Fahad Alsharari, Faruk Karaaslan, Aleksandra Fedajev, Daniela Gîfu, Hina Gulzar, Haitham A. El-Ghareeb, Masooma Raza Hashmi, Hewayda El-Ghawalby, Hoang Viet Long, Le Hoang Son, F. Nirmala Irudayam, Branislav Ivanov, S. Jafari, Jeong Gon Lee, Milena Jevtić, Sudan Jha, Junhui Kim, Ilanthenral Kandasamy, W.B. Vasantha Kandasamy, Darjan Karabašević, Songül Karabatak, Abdullah Kargın, M. Karthika, Ieva Meidute-Kavaliauskiene, Madad Khan, Majid Khan, Manju Khari, Kifayat Ullah, K. Kishore, Kul Hur, Santanu Kumar Patro, Prem Kumar Singh, Raghvendra Kumar, Tapan Kumar Roy, Malayalan Lathamaheswari, Luu Quoc Dat, T. Madhumathi, Tahir Mahmood, Mladjan Maksimovic, Gunasekaran Manogaran, Nivetha Martin, M. Kasi Mayan, Mai Mohamed, Mohamed Talea, Muhammad Akram, Muhammad Gulistan, Raja Muhammad Hashim, Muhammad Riaz, Muhammad Saeed, Rana Muhammad Zulqarnain, Nada A. Nabeeh, Deivanayagampillai Nagarajan, Xenia Negrea, Nguyen Xuan Thao, Jagan M. Obbineni, Angelo de Oliveira, M. Parimala, Gabrijela Popovic, Ishaani Priyadarshini, Yaser Saber, Mehmet Șahin, Said Broumi, A. A. Salama, M. Saleh, Ganeshsree Selvachandran, Dönüș Șengür, Shio Gai Quek, Songtao Shao, Dragiša Stanujkić, Surapati Pramanik, Swathi Sundari Sundaramoorthy, Mirela Teodorescu, Selçuk Topal, Muhammed Turhan, Alptekin Ulutaș, Luige Vlădăreanu, Victor Vlădăreanu, Ştefan Vlăduţescu, Dan Valeriu Voinea, Volkan Duran, Navneet Yadav, Yanhui Guo, Naveed Yaqoob, Yongquan Zhou, Young Bae Jun, Xiaohong Zhang, Xiao Long Xin, Edmundas Kazimieras Zavadskas
Computability Theory (hybrid meeting)
Over the last decade computability theory has seen many new and
fascinating developments that have linked the subject much closer
to other mathematical disciplines inside and outside of logic.
This includes, for instance, work on enumeration degrees that
has revealed deep and surprising relations to general topology,
the work on algorithmic randomness that is closely tied to
symbolic dynamics and geometric measure theory.
Inside logic there are connections to model theory, set theory, effective descriptive
set theory, computable analysis and reverse mathematics.
In some of these cases the bridges to seemingly distant mathematical fields
have yielded completely new proofs or even solutions of open problems
in the respective fields. Thus, over the last decade, computability theory
has formed vibrant and beneficial interactions with other mathematical
fields.
The goal of this workshop was to bring together researchers representing
different aspects of computability theory to discuss recent advances, and to
stimulate future work
A journey through computability, topology and analysis
This thesis is devoted to the exploration of the complexity of some mathematical problems using the framework of computable analysis and descriptive set theory. We will especially focus on Weihrauch reducibility, as a means to compare the uniform computational strength of problems. After a short introduction of the relevant background notions, we investigate the uniform computational content of the open and clopen Ramsey theorems. In particular, since there is not a canonical way to phrase these theorems as multi-valued functions, we identify 8 different multi-valued functions (5 corresponding to the open Ramsey theorem and 3 corresponding to the clopen Ramsey theorem) and study their degree from the point of view of Weihrauch, strong Weihrauch and arithmetic Weihrauch reducibility. We then discuss some new operators on multi-valued functions and study their algebraic properties and the relations with other previously studied operators on problems. These notions turn out to be extremely relevant when exploring the Weihrauch degree of the problem DS of computing descending sequences in ill-founded linear orders. They allow us to show that DS, and the Weihrauch equivalent problem BS of finding bad sequences through non-well quasi-orders, while being very "hard" to solve, are rather weak in terms of uniform computational strength. We then generalize DS and BS by considering Gamma-presented orders, where Gamma is a Borel pointclass or Delta11, Sigma11, Pi11. We study the obtained DS-hierarchy and BS-hierarchy of problems in comparison with the (effective) Baire hierarchy and show that they do not collapse at any finite level. Finally, we focus on the characterization, from the point of view of descriptive set theory, of some conditions involving the notions of Hausdorff/Fourier dimension and of Salem sets. We first work in the hyperspace K([0,1]) of compact subsets of [0,1] and show that the closed Salem sets form a Pi03-complete family. This is done by characterizing the complexity of the family of sets having sufficiently large Hausdorff or Fourier dimension. We also show that the complexity does not change if we increase the dimension of the ambient space and work in K([0,1]^d). We also generalize the results by relaxing the compactness of the ambient space, and show that the closed Salem sets are still Pi03-complete when we endow K(R^d) with the Fell topology. A similar result holds also for the Vietoris topology. We conclude by showing how these results can be used to characterize the Weihrauch degree of the functions computing the Hausdorff and Fourier dimensions
On cohesive powers of linear orders
Cohesive powers of computable structures are effective analogs of
ultrapowers, where cohesive sets play the role of ultrafilters. Let ,
, and denote the respective order-types of the natural numbers,
the integers, and the rationals when thought of as linear orders. We
investigate the cohesive powers of computable linear orders, with special
emphasis on computable copies of . If is a computable
copy of that is computably isomorphic to the standard presentation of
, then every cohesive power of has order-type . However, there are computable copies of , necessarily not
computably isomorphic to the standard presentation, having cohesive powers not
elementarily equivalent to . For example, we show that
there is a computable copy of with a cohesive power of order-type
. Our most general result is that if is either a set or a set, thought of as a
set of finite order-types, then there is a computable copy of with a
cohesive power of order-type , where denotes
the shuffle of the order-types in and the order-type . Furthermore, if is finite and non-empty, then there is a
computable copy of with a cohesive power of order-type
- …