Fuzzy Conifold YF6Y_F^6 and Monopoles on SF2×SF2S_F^2\times S_F^2

In this article, we construct the fuzzy (finite dimensional) analogues of the conifold $Y^6$ and its base $X^5$. We show that fuzzy $X^5$ is (the analogue of) a principal U(1) bundle over fuzzy spheres $S^2_F \times S^2_F$ and explicitly construct the associated monopole bundles. In particular our construction provides an explicit discretization of the spaces $T^{\kappa,\kappa}$ and $T^{\kappa,0}$

The Importance of Forgetting: Limiting Memory Improves Recovery of Topological Characteristics from Neural Data

We develop of a line of work initiated by Curto and Itskov towards understanding the amount of information contained in the spike trains of hippocampal place cells via topology considerations. Previously, it was established that simply knowing which groups of place cells fire together in an animal's hippocampus is sufficient to extract the global topology of the animal's physical environment. We model a system where collections of place cells group and ungroup according to short-term plasticity rules. In particular, we obtain the surprising result that in experiments with spurious firing, the accuracy of the extracted topological information decreases with the persistence (beyond a certain regime) of the cell groups. This suggests that synaptic transience, or forgetting, is a mechanism by which the brain counteracts the effects of spurious place cell activity

Arguments Whose Strength Depends on Continuous Variation

Both the traditional Aristotelian and modern symbolic approaches to logic have seen logic in terms of discrete symbol processing. Yet there are several kinds of argument whose validity depends on some topological notion of continuous variation, which is not well captured by discrete symbols. Examples include extrapolation and slippery slope arguments, sorites, fuzzy logic, and those involving closeness of possible worlds. It is argued that the natural first attempts to analyze these notions and explain their relation to reasoning fail, so that ignorance of their nature is profound

Tensor model and dynamical generation of commutative nonassociative fuzzy spaces

Rank-three tensor model may be regarded as theory of dynamical fuzzy spaces, because a fuzzy space is defined by a three-index coefficient of the product between functions on it, f_a*f_b=C_ab^cf_c. In this paper, this previous proposal is applied to dynamical generation of commutative nonassociative fuzzy spaces. It is numerically shown that fuzzy flat torus and fuzzy spheres of various dimensions are classical solutions of the rank-three tensor model. Since these solutions are obtained for the same coupling constants of the tensor model, the cosmological constant and the dimensions are not fundamental but can be regarded as dynamical quantities. The symmetry of the model under the general linear transformation can be identified with a fuzzy analog of the general coordinate transformation symmetry in general relativity. This symmetry of the tensor model is broken at the classical solutions. This feature may make the model to be a concrete finite setting for applying the old idea of obtaining gravity as Nambu-Goldstone fields of the spontaneous breaking of the local translational symmetry.Comment: Adding discussions on effective geometry, a note added, four references added, other minor changes, 27 pages, 17 figure

Spectral geometry with a cut-off: topological and metric aspects

Inspired by regularization in quantum field theory, we study topological and metric properties of spaces in which a cut-off is introduced. We work in the framework of noncommutative geometry, and focus on Connes distance associated to a spectral triple (A, H, D). A high momentum (short distance) cut-off is implemented by the action of a projection P on the Dirac operator D and/or on the algebra A. This action induces two new distances. We individuate conditions making them equivalent to the original distance. We also study the Gromov-Hausdorff limit of the set of truncated states, first for compact quantum metric spaces in the sense of Rieffel, then for arbitrary spectral triples. To this aim, we introduce a notion of "state with finite moment of order 1" for noncommutative algebras. We then focus on the commutative case, and show that the cut-off induces a minimal length between points, which is infinite if P has finite rank. When P is a spectral projection of $D$, we work out an approximation of points by non-pure states that are at finite distance from each other. On the circle, such approximations are given by Fejer probability distributions. Finally we apply the results to Moyal plane and the fuzzy sphere, obtained as Berezin quantization of the plane and the sphere respectively.Comment: Reference added. Minor corrections. Published version. 38 pages, 2 figures. Journal of Geometry and Physics 201

Introduction to Gestural Similarity in Music. An Application of Category Theory to the Orchestra

Mathematics, and more generally computational sciences, intervene in several aspects of music. Mathematics describes the acoustics of the sounds giving formal tools to physics, and the matter of music itself in terms of compositional structures and strategies. Mathematics can also be applied to the entire making of music, from the score to the performance, connecting compositional structures to acoustical reality of sounds. Moreover, the precise concept of gesture has a decisive role in understanding musical performance. In this paper, we apply some concepts of category theory to compare gestures of orchestral musicians, and to investigate the relationship between orchestra and conductor, as well as between listeners and conductor/orchestra. To this aim, we will introduce the concept of gestural similarity. The mathematical tools used can be applied to gesture classification, and to interdisciplinary comparisons between music and visual arts.Comment: The final version of this paper has been published by the Journal of Mathematics and Musi

Toward a multilevel representation of protein molecules: comparative approaches to the aggregation/folding propensity problem

This paper builds upon the fundamental work of Niwa et al. [34], which provides the unique possibility to analyze the relative aggregation/folding propensity of the elements of the entire Escherichia coli (E. coli) proteome in a cell-free standardized microenvironment. The hardness of the problem comes from the superposition between the driving forces of intra- and inter-molecule interactions and it is mirrored by the evidences of shift from folding to aggregation phenotypes by single-point mutations [10]. Here we apply several state-of-the-art classification methods coming from the field of structural pattern recognition, with the aim to compare different representations of the same proteins gathered from the Niwa et al. data base; such representations include sequences and labeled (contact) graphs enriched with chemico-physical attributes. By this comparison, we are able to identify also some interesting general properties of proteins. Notably, (i) we suggest a threshold around 250 residues discriminating "easily foldable" from "hardly foldable" molecules consistent with other independent experiments, and (ii) we highlight the relevance of contact graph spectra for folding behavior discrimination and characterization of the E. coli solubility data. The soundness of the experimental results presented in this paper is proved by the statistically relevant relationships discovered among the chemico-physical description of proteins and the developed cost matrix of substitution used in the various discrimination systems.Comment: 17 pages, 3 figures, 46 reference