Towards Practical Typechecking for Macro Tree Transducers

Macro tree transducers (mtt) are an important model that both covers many useful XML transformations and allows decidable exact typechecking. This paper reports our first step toward an implementation of mtt typechecker that has a practical efficiency. Our approach is to represent an input type obtained from a backward inference as an alternating tree automaton, in a style similar to Tozawa's XSLT0 typechecking. In this approach, typechecking reduces to checking emptiness of an alternating tree automaton. We propose several optimizations (Cartesian factorization, state partitioning) on the backward inference process in order to produce much smaller alternating tree automata than the naive algorithm, and we present our efficient algorithm for checking emptiness of alternating tree automata, where we exploit the explicit representation of alternation for local optimizations. Our preliminary experiments confirm that our algorithm has a practical performance that can typecheck simple transformations with respect to the full XHTML in a reasonable time

How to Design Non-Kekule Polyhex Graphs?

An efficient algorithm is proposed for designing alternant hydrocarbons with the same numbers of starred and unstarred carbon atoms but with no Kekule structure. This algorithm is based on the interesting alternant properties of the non-zero coefficients of the NBMO (non-bonding molecular orbital) of special series of odd alternant hydrocarbon radicals

A mixture of sparse coding models explaining properties of face neurons related to holistic and parts-based processing

Experimental studies have revealed evidence of both parts-based and holistic representations of objects and faces in the primate visual system. However, it is still a mystery how such seemingly contradictory types of processing can coexist within a single system. Here, we propose a novel theory called mixture of sparse coding models, inspired by the formation of category-specific subregions in the inferotemporal (IT) cortex. We developed a hierarchical network that constructed a mixture of two sparse coding submodels on top of a simple Gabor analysis. The submodels were each trained with face or non-face object images, which resulted in separate representations of facial parts and object parts. Importantly, evoked neural activities were modeled by Bayesian inference, which had a top-down explaining-away effect that enabled recognition of an individual part to depend strongly on the category of the whole input. We show that this explaining-away effect was indeed crucial for the units in the face submodel to exhibit significant selectivity to face images over object images in a similar way to actual face-selective neurons in the macaque IT cortex. Furthermore, the model explained, qualitatively and quantitatively, several tuning properties to facial features found in the middle patch of face processing in IT as documented by Freiwald, Tsao, and Livingstone (2009). These included, in particular, tuning to only a small number of facial features that were often related to geometrically large parts like face outline and hair, preference and anti-preference of extreme facial features (e.g., very large/small inter-eye distance), and reduction of the gain of feature tuning for partial face stimuli compared to whole face stimuli. Thus, we hypothesize that the coding principle of facial features in the middle patch of face processing in the macaque IT cortex may be closely related to mixture of sparse coding models.Peer reviewe

Distance Polynomial and the Related Counting Polynomials

Interesting mathematical properties of the distance polynomial SG(x) proposed by the present author in 1973 are reintroduced together with several new findings. Although many results are given in the form of “Theorems”, most of them have not rigorously been proved mathematically but readers are challenged to provide their proofs. (doi: 10.5562/cca2311

Mathematical Meaning and Importance of the Topological Index Z

The role of the topological index, Z G, proposed by the present author in 1971, in various problems and topics in elementary mathematics is introduced, namely, (i) Pascal’s and asymmetrical Pascal’s triangle, (ii) Fibonacci, Lucas, and Pell numbers, (iii) Pell equation, (iv) Pythagorean, Heronian, and Eisenstein triangles. It is shown that all the algebras in these problems can be easily obtained, graph-theoretically interpreted, and systematically related with each other by introducing certain series of graphs whose Z G values represent the series of numbers involved therein. Finally, an ambitious conjecture is proposed: for any recursive relation of the type of Fibonacci numbers, there always exist a series of graphs whose Z-indices obey the same recursive relation. Important role of Z G in algebraic number theory is also discussed

Chemistry-Relevant Isospectral Graphs. Acyclic Conjugated Polyenes

By using the topological index Z and Z-counting polynomial proposed by the present author isospectral (IS) pairs of acyclic conjugated polyenes (C2nH2n+2) were studied. Besides the hitherto known smallest pair of n = 6, four and twenty seven pairs of n = 7 and 8, respectively, were first reported. Inspection of these results revealed several new features of IS tree graphs, i.e., appearance of two pairs of endospectral vertices in a tree graph and existence of several families of IS pair graphs whose Z-indices systematically grow up to infinity. Further several IS pairs were found to be closely related with each other topologically and called “intrinsic” IS pairs. Important role of the Z-index for analyzing the IS graphs is demonstrated. This work is licensed under a Creative Commons Attribution 4.0 International License

Convex Configurations on Nana-kin-san Puzzle

We investigate a silhouette puzzle that is recently developed based on the golden ratio. Traditional silhouette puzzles are based on a simple tile. For example, the tangram is based on isosceles right triangles; that is, each of seven pieces is formed by gluing some identical isosceles right triangles. Using the property, we can analyze it by hand, that is, without computer. On the other hand, if each piece has no special property, it is quite hard even using computer since we have to handle real numbers without numerical errors during computation. The new silhouette puzzle is between them; each of seven pieces is not based on integer length and right angles, but based on golden ratio, which admits us to represent these seven pieces in some nontrivial way. Based on the property, we develop an algorithm to handle the puzzle, and our algorithm succeeded to enumerate all convex shapes that can be made by the puzzle pieces. It is known that the tangram and another classic silhouette puzzle known as Sei-shonagon chie no ita can form 13 and 16 convex shapes, respectively. The new puzzle, Nana-kin-san puzzle, admits to form 62 different convex shapes

A New View of Hybridized Atomic Orbitals from N-dimensional World

By taking a birds-eye view from the n-dimensional world (or n-space), it was found that thež conventionally used sp, sp2, and sp3 hybridized atomic orbitals belong to the spn hybridization and their geometrical shapes correspond to n-simplexes, which are, respectively, the smallest geometrical objects in n-space. Similarly, sp, sp2d, and sp2d3 hybridizations are found to belong to the spndn–1 hybridization, whose geometrical shapes correspond to n-cross polytopes (obtained from 2n vertices equidistantly located on n rectangular coordinate axes). Another series of n-cube hybridization is also discussed, whose 3-space member is 8-coordinated cubic hybridization, sp3d3f. General analytical forms of the wavefunctions of these three series of hybridized atomic orbitals in n-space are obtained. The periodic table and related problems of atoms and molecules in n-space are discussed