873 research outputs found
Most undirected random graphs are amplifiers of selection for Birth-death dynamics, but suppressors of selection for death-Birth dynamics
We analyze evolutionary dynamics on graphs, where the nodes represent
individuals of a population. The links of a node describe which other
individuals can be displaced by the offspring of the individual on that node.
Amplifiers of selection are graphs for which the fixation probability is
increased for advantageous mutants and decreased for disadvantageous mutants. A
few examples of such amplifiers have been developed, but so far it is unclear
how many such structures exist and how to construct them. Here, we show that
almost any undirected random graph is an amplifier of selection for Birth-death
updating, where an individual is selected to reproduce with probability
proportional to its fitness and one of its neighbors is replaced by that
offspring at random. If we instead focus on death-Birth updating, in which a
random individual is removed and its neighbors compete for the empty spot, then
the same ensemble of graphs consists of almost only suppressors of selection
for which the fixation probability is decreased for advantageous mutants and
increased for disadvantageous mutants. Thus, the impact of population structure
on evolutionary dynamics is a subtle issue that will depend on seemingly minor
details of the underlying evolutionary process
Spectral and wave function statistics in Quantum digraphs
This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.Spectral and wave function statistics of the quantum directed graph, QdG, are studied. The crucial feature of this model is that the direction of a bond (arc) corresponds to the direction of the waves propagating along it. We pay special attention to the full Neumann digraph, FNdG, which consists of pairs of antiparallel arcs between every node, and differs from the full Neumann graph, FNG, in that the two arcs have two incommensurate lengths. The spectral statistics of the FNG (with incommensurate bond lengths) is believed to be universal, i.e. to agree with that of the random matrix theory, RMT, in the limit of large graph size. However, the standard perturbative treatment of the field theoretical representation of the 2-point correlation function [1, 2] for a FNG, does not account for this behaviour. The nearest-neighbor spacing distribution of the closely related FNdG is studied numerically. An original, efficient algorithm for the generation of the spectrum of large graphs allows for the observation that the distribution approaches indeed universality at increasing graph size (although the convergence cannot be ascertained), in particular "level repulsion" is confirmed. The numerical technique employs a new secular equation which generalizes the analogous object known for undirected graphs [3, 4], and is based on an adaptation to digraphs of the idea of wave function continuity. In view of the contradiction between the field theory [2] and the strong indications of universality, a non-perturbative approach to analysing the universal limit is presented. The substitution of the FNG by the FNdG results in a field theory with fewer degrees of freedom. Despite this simplification, the attempt is inconclusive. Possible applications of this approach are suggested.
Regarding the wave function statistics, a field theoretical representation for the
spectral average of the wave intensity on an fixed arc is derived and studied in the universal limit. The procedure originates from the study of wave function statistics on disordered metallic grains [5] and is used in conjunction with the field theory approach pioneered in [2]
The combinatorics of minimal unsatisfiability: connecting to graph theory
Minimally Unsatisfiable CNFs (MUs) are unsatisfiable CNFs where removing any clause destroys unsatisfiability. MUs are the building blocks of unsatisfia-bility, and our understanding of them can be very helpful in answering various algorithmic and structural questions relating to unsatisfiability. In this thesis we study MUs from a combinatorial point of view, with the aim of extending the understanding of the structure of MUs. We show that some important classes of MUs are very closely related to known classes of digraphs, and using arguments from logic and graph theory we characterise these MUs.Two main concepts in this thesis are isomorphism of CNFs and the implica-tion digraph of 2-CNFs (at most two literals per disjunction). Isomorphism of CNFs involves renaming the variables, and flipping the literals. The implication digraph of a 2-CNF F has both arcs (¬a → b) and (¬b → a) for every binary clause (a ∨ b) in F .In the first part we introduce a novel connection between MUs and Minimal Strong Digraphs (MSDs), strongly connected digraphs, where removing any arc destroys the strong connectedness. We introduce the new class DFM of special MUs, which are in close correspondence to MSDs. The known relation between 2-CNFs and implication digraphs is used, but in a simpler and more direct way, namely that we have a canonical choice of one of the two arcs. As an application of this new framework we provide short and intuitive new proofs for two im-portant but isolated characterisations for nonsingular MUs (every literal occurs at least twice), both with ingenious but complicated proofs: Characterising 2-MUs (minimally unsatisfiable 2-CNFs), and characterising MUs with deficiency 2 (two more clauses than variables).In the second part, we provide a fundamental addition to the study of 2-CNFs which have efficient algorithms for many interesting problems, namely that we provide a full classification of 2-MUs and a polytime isomorphism de-cision of this class. We show that implication digraphs of 2-MUs are “Weak Double Cycles” (WDCs), big cycles of small cycles (with possible overlaps). Combining logical and graph-theoretical methods, we prove that WDCs have at most one skew-symmetry (a self-inverse fixed-point free anti-symmetry, re-versing the direction of arcs). It follows that the isomorphisms between 2-MUs are exactly the isomorphisms between their implication digraphs (since digraphs with given skew-symmetry are the same as 2-CNFs). This reduces the classifi-cation of 2-MUs to the classification of a nice class of digraphs.Finally in the outlook we discuss further applications, including an alter-native framework for enumerating some special Minimally Unsatisfiable Sub-clause-sets (MUSs)
Matter-antimatter asymmetry restrains the dimensionality of neural representations: quantum decryption of large-scale neural coding
Projections from the study of the human universe onto the study of the
self-organizing brain are herein leveraged to address certain concerns raised
in latest neuroscience research, namely (i) the extent to which neural codes
are multidimensional; (ii) the functional role of neural dark matter; (iii) the
challenge to traditional model frameworks posed by the needs for accurate
interpretation of large-scale neural recordings linking brain and behavior. On
the grounds of (hyper-)self-duality under (hyper-)mirror supersymmetry,
inter-relativistic quantum principles are introduced, whose consolidation, as
spin-geometrical pillars of a network- and game-theoretical construction, is
conducive to (i) the high-precision reproduction and reinterpretation of core
experimental observations on neural coding in the self-organizing brain, with
the instantaneous geometric dimensionality of neural representations of a
spontaneous behavioral state being proven to be at most 16, unidirectionally;
(ii) a possible role for spinor (co-)representations, as the latent building
blocks of self-organizing cortical circuits subserving (co-)behavioral states;
(iii) an early crystallization of pertinent multidimensional synaptic
(co-)architectures, whereby Lorentz (co-)partitions are in principle
verifiable; and, ultimately, (iv) potentially inverse insights into
matter-antimatter asymmetry. New avenues for the decryption of large-scale
neural coding in health and disease are being discussed.Comment: 33 pages;3 figures;1 table;minor edit
Computation in Finitary Stochastic and Quantum Processes
We introduce stochastic and quantum finite-state transducers as
computation-theoretic models of classical stochastic and quantum finitary
processes. Formal process languages, representing the distribution over a
process's behaviors, are recognized and generated by suitable specializations.
We characterize and compare deterministic and nondeterministic versions,
summarizing their relative computational power in a hierarchy of finitary
process languages. Quantum finite-state transducers and generators are a first
step toward a computation-theoretic analysis of individual, repeatedly measured
quantum dynamical systems. They are explored via several physical systems,
including an iterated beam splitter, an atom in a magnetic field, and atoms in
an ion trap--a special case of which implements the Deutsch quantum algorithm.
We show that these systems' behaviors, and so their information processing
capacity, depends sensitively on the measurement protocol.Comment: 25 pages, 16 figures, 1 table; http://cse.ucdavis.edu/~cmg; numerous
corrections and update
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A Comparative Study Of Directional Connections In Popular U.S. And Chinese High School Mathematics Textbook Problems
Mathematical connection has received increasing attention and become one major goal in mathematics education. Two types of connections are distinguished: (a) between-concept connection, which cuts across two concepts; and (b) within-concept connection, which links two representations of one concept. For example, from the theoretical probability to experimental probability is a between-concept connection; generate a graph of a circle from its equation is a within-concept connection. Based on the directionality, unidirectional and bidirectional connections are discerned. Bidirectional connection portrays a pair of a typical and a reverse connection. The benefits of connections, especially bidirectional connections, are widely endorsed. However, researchers indicated that students and even teachers usually make unidirectional connections, and underlying reasons may be the curriculum and cognitive aspects. Previous studies have reported differences in learning opportunities for bidirectional connections in U.S. and Chinese textbook problems, but few have explored the high school level.
This study addressed this issue by comparing the directionality of mathematical connections and textbook-problem features in popular U.S. (the UCSMP series) and Chinese (the PEP-A series) high school mathematics textbook problems. The results indicated that the between-concept condition and unidirectional connections dominated textbook problems. Mathematical topic, contextual feature, and visual feature were most likely to contribute to different conditions of connections. Overall, problems dealing with quadratic relations from Chinese textbooks presented a vigorous network of more unique and total between-concept connections with balanced typical and reverse directions than the U.S. counterparts. Problems from U.S. textbooks showed a denser network of (a) within-concept connections in two topics and (b) between-concept connections in probability and combinatorics than the Chinese counterparts, but still exhibited an emphasis on specific concepts, representations, and directionality. The study reached a generalized statement that the new-to-prior knowledge direction was largely overlooked in textbook problems. The results have implications for adopting graph theory and Social Network Analysis to visualize and evaluate mathematical connections and informing mathematics teachers and textbook authors to pay attention to the new-to-prior knowledge connection
Fault propagation, detection and analysis in process systems
Process systems are often complicated and liable to experience faults and their effects. Faults can adversely affect the safety of the plant, its environmental impact and economic operation. As such, fault diagnosis in process systems is an active area of research and development in both academia and
industry.
The work reported in this thesis contributes to fault diagnosis by exploring the modelling and
analysis of fault propagation and detection in process systems. This is done by posing and answering three research questions. What are the necessary ingredients of a fault diagnosis model? What information should a fault diagnosis model yield? Finally, what types of model are appropriate to fault diagnosis?
To answer these questions , the assumption of the research is that the behaviour of a process system arises from the causal structure of the process system. On this basis, the research presented in this thesis develops a two-level approach to fault diagnosis based on detailed process information, and modelling and analysis techniques for representing causality.
In the first instance, a qualitative approach is developed called a level 1 fusion. The level 1 fusion models the detailed causality of the system using digraphs. The level 1 fusion is a causal map of the process. Such causal maps can be searched to discover and analyse fault propagation paths through the process.
By directly building on the level 1 fusion, a quantitative level 2 fusion is developed which uses a type of digraph called a Bayesian network. By associating process variables with fault variables, and using conditional probability theory, it is shown how measured effects can be used to calculate and
rank the probability of candidate causes.
The novel contributions are the development of a systematic approach to fault diagnosis based on modelling the chemistry, physics, and architecture of the process. It is also shown how the control and instrumentation system constrains the casualty of the process. By demonstrating how digraph models
can be reversed, it is shown how both cause-to-effect and effect-to-cause analysis can be carried out.
In answering the three research questions, this research shows that it is feasible to gain detailed insights into fault propagation by qualitatively modelling the physical causality of the process system. It is also shown that a qualitative fault diagnosis model can be used as the basis for a quantitative fault
diagnosis modelOpen Acces
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