Two-state number-conserving cellular automaton (NCCA) is a cellular automaton of which cell states are 0 or 1, and the total sum of all the states of cells is kept for any time step. It is a kind of ...particle-based modeling of physical systems. We introduce a new structure of its value-1 patterns, which we call a “bundle pair” and a “bundle quad”. By employing this structure, we show a relation between the neighborhood size n and n - 2 NCCAs.
The present article complements the earlier ones in this series. The first part contains various results on the constituent system Cκ(M) of a graph model M, and on its feasibility system Iη(M) (which ...comprises a number of identities that define the number conservation rules). Those results include the general form of the (particle) number conservation rules in models without explicit propagator mixing.
A few types of graph models (including self-conjugate and quasi-normal models) are defined. Oversimplifying, a model M is self-conjugate if it has a certain (weak) combinatorial type of C-symmetry, and M is quasi-normal if Iη(M) fully determines for which multisets of coloured, unlinked half-edges there is a graph in M with that exact multiset. It is shown not only that every self-conjugate model is quasi-normal, but also that some extensions of self-conjugate models are still quasi-normal. Most conveniently, self-conjugate models (and even some extended models) may be recognized in polynomial time.
These results seem to lead to the following conclusion: for many (possibly ‘most’ of the) consistent, relevant QFT models, a complete correlation function 〈F〉 is graphical — i.e. there exist Feynman diagram(s) for 〈F〉 — if and only if 〈F〉 is allowed by the number conservation rules (i.e. by a complete system of such rules). Since these rules can be computed in polynomial time then, for many QFT models, deciding whether 〈F〉 is graphical also takes polynomial time.
We present a simple approach for finding all number-conserving two-dimensional cellular automata with the von Neumann neighborhood. We demonstrate the efficiency of this approach by enumerating and ...describing four special cases: totalistic, outer-totalistic, binary and three-state cellular automata. The last result was not published before. We then proceed to find all reversible two-dimensional number-conserving cellular automata with three states with the von Neumann neighborhood and show there are only trivial ones.
Most of the discussion in the literature of the Majorana fermions (M.F.’s) believed to exist in so-called 2D
p
+
ip
Fermi superfluids, and in particular of their possible application in topological ...quantum computing (TQC) has analyzed the problem using the familiar Bogoliubov–de Gennes (mean-field) formalism, which conserves total electron number only mod 2. We raise the question: Suppose that we require that electron number be conserved, period, at all stages of the calculation, then what are the consequences for the characteristic properties of M.F.’s, in particular for their (physical) braiding statistics? While we do not provide a definitive answer to this question, our considerations suggest that these consequences could be enough to destroy the possibility of using these excitations for TQC, and certainly indicate that the problem deserves further study.
Little is known about the dynamics of
k
-ary (binary, ternary, quaternary, quinary, etc.) reversible number-conserving cellular automata. Here, we present some preliminary results in the case of ...seven states. In particular, we examine one of the most complex seven-state reversible and number-conserving rules and provide a full description of its dynamics.
Introduction: Subitizing, a quick apprehension of the numerosity of a small set of items, is consistently utilized to support early number understanding. Perceptual subitizing is the innate ability ...to recognize less than five items without consciously using other mental or mathematical processes. Conceptual subitizing, which requires higher-level abilities, means perceiving the quantities as groups and performing a mental process on them. Research on conceptual and perceptual subitizing indicates some limitations about the activities regarding the children's early number development. So, MacDonald and Wilkins (2016) developed a framework that explained the types of activities that young children used while subitizing. In this framework, five sets of perceptual subitizing activity were described to explain how young children's perceptual subitizing activity changed. Besides, two types of conceptual subitizing activities were defined to explain how children's limited or flexible number understanding related to their subitizing activity. These seven different types of activities characterize the changes in children's subitizing actions. The study aims to investigate the relationship between children's number understanding and subitizing activity. Methods: A teaching experiment was conducted with two preschool-aged children to analyze what perceptual and conceptual processes children relied upon when subitizing. The teaching experiment consisted of twenty-six sessions. The interviews were conducted to determine whether children are able to conserve numbers or not, and whether they rely on a variety of different types of subitizing activity or not. After the interviews, 26 teaching sessions were carried out with two preconserver children. Results: In the experimental process, it was observed that the children rely on the color of items, the gap between items, and symmetrical aspects of items when perceptually subitizing. However, they could not manage to transition their subitizing activity from perceptual to conceptual subitizing. The study indicates that children's subitizing skills were closely related to their number conservation development. Discussion: Based on the findings from this study, for Eren and Beren, subitizing activities were found to be perceptually limited. Specifically, it was found that four types of perceptual subitizing emerged to explain how symmetry, the gap between items, color of items, and canonical patterns promoted strategies that children relied on when constructing number understanding. During the teaching experiment, although these children carried out the activities that required the separating and combining numbers and seeing the relationship between the subgroups and the composite groups, they used perceptual units in this process. The relationship between the number conservation activity and the conceptual subitizing activity requires the coordination of thinking structures related to both ordering and classification. However, it was found that the children could not move from perceptual to the conceptual subitizing. Limitations: As all studies have some limitations, this study has, too. One of the limitations of the study is the sample size/number of participants. But teaching experiments aim to get a deep understanding, studying with a small sample is an obligation. Secondly, this study focused on some compounds of subitizing such as perceptual and conceptual ones. Conclusion: In order to make the transition from perceptual subitizing to conceptual subitizing children should have more experiences with subitizing activities.When designing mathematical games and assessments for young children, being aware of different types of subitizing categories may provide better support children's number understanding and subitizing.
The eigenvectors of the particle number operator in second quantization are characterized by the block sparsity of their matrix product state representations. This is shown to generalize to other ...classes of operators. Imposing block sparsity yields a scheme for conserving the particle number that is commonly used in applications in physics. Operations on such block structures, their rank truncation, and implications for numerical algorithms are discussed. Explicit and rank-reduced matrix product operator representations of one- and two-particle operators are constructed that operate only on the non-zero blocks of matrix product states.
Extending the basic formalism described in a closely related article, this paper defines the reduced equations of a graph model and presents various theoretical results involving those equations, ...both diophantine and modular. The reduced equations contribute to clarify the existence (or non-existence) of certain types of identities, for some classes of models (a number of which is analysed in the second part of this paper).
It is also pointed out that there exist systems of equations (related to the reduced equations) that can replace the system of constituent equations for the purpose of deriving the identities of a given model. A definite diophantine system is proposed, one for which the coefficient matrix is typically smaller than that of the system of constituent equations.
In the context of Quantum Field Theory (QFT) there is often the need to find sets of graph-like diagrams (the so-called Feynman diagrams) for a given physical model. If negative, the answer to the ...related problem ‘Are there any diagrams with this set of external fields?’ may settle certain physical questions at once. Here the latter problem is formulated in terms of a system of linear diophantine equations derived from the Lagrangian density, from which necessary conditions for the existence of the required diagrams may be obtained. Those conditions are equalities that look like either linear diophantine equations or linear modular (i.e. congruence) equations, and may be found by means of fairly simple algorithms that involve integer computations. The diophantine equations so obtained represent (particle) number conservation rules, and are related to the conserved (additive) quantum numbers that may be assigned to the fields of the model.
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