Algebraic Properties of Generalized Graph Laplacians: Resistor Networks, Critical Groups, and Homological Algebra

We propose an algebraic framework for generalized graph Laplacians which unifies the study of resistor networks, the critical group, and the eigenvalues of the Laplacian and adjacency matrices. Given a graph with boundary $G$ together with a generalized Laplacian $L$ with entries in a commutative ring $R$, we define a generalized critical group $\Upsilon_R(G,L)$. We relate $\Upsilon_R(G,L)$ to spaces of harmonic functions on the network using the Hom, Tor, and Ext functors of homological algebra. We study how these algebraic objects transform under combinatorial operations on the network $(G,L)$, including harmonic morphisms, layer-stripping, duality, and symmetry. In particular, we use layer-stripping operations from the theory of resistor networks to systematize discrete harmonic continuation. This leads to an algebraic characterization of the graphs with boundary that can be completely layer-stripped, an algorithm for simplifying computation of $\Upsilon_R(G,L)$, and upper bounds for the number of invariant factors in the critical group and the multiplicity of Laplacian eigenvalues in terms of geometric quantities.Comment: 73 pages, 27 figures. This revised version submitted to SIDMA on Jan. 9 201

Social approach in preschool children with Williams syndrome: The role of the face.

Background Indiscriminate social approach behaviour is a salient aspect of the Williams syndrome (WS) behavioural phenotype. The present study examines approach behaviour in preschoolers with WS and evaluates the role of the face in WS social approach behaviour. Method Ten preschoolers with WS (aged 3-6 years) and two groups of typically developing children, matched to the WS group on chronological or mental age, participated in an observed play session. The play session incorporated social and non-social components including two components that assessed approach behaviour towards strangers, one in which the stranger’s face could be seen and one in which the stranger’s face was covered. Results In response to the non-social aspects of the play session, the WS group behaved similarly to both control groups. In contrast, the preschoolers with WS were significantly more willing than either control group to engage with a stranger, even when the stranger’s face could not be seen. Conclusion The findings challenge the hypothesis that an unusual attraction to the face directly motivates social approach behaviour in individuals with WS

Topological Observables in Semiclassical Field Theories

We give a geometrical set up for the semiclassical approximation to euclidean field theories having families of minima (instantons) parametrized by suitable moduli spaces ${\cal M}$. The standard examples are of course Yang-Mills theory and non-linear $\sigma$-models. The relevant space here is a family of measure spaces \tilde {\cal N} \ra {\cal M}, with standard fibre a distribution space, given by a suitable extension of the normal bundle to ${\cal M}$ in the space of smooth fields. Over $\tilde {\cal N}$ there is a probability measure $d\mu$ given by the twisted product of the (normalized) volume element on ${\cal M}$ and the family of gaussian measures with covariance given by the tree propagator $C_\phi$ in the background of an instanton $\phi \in {\cal M}$. The space of ``observables", i.e. measurable functions on ($\tilde {\cal N}, \, d\mu$), is studied and it is shown to contain a topological sector, corresponding to the intersection theory on ${\cal M}$. The expectation value of these topological ``observables" does not depend on the covariance; it is therefore exact at all orders in perturbation theory and can moreover be computed in the topological regime by setting the covariance to zero.Comment: 11 page

Metabolically exaggerated cardiac reactions to acute psychological stress revisited

The reactivity hypothesis postulates that large magnitude cardiovascular reactions to psychological stress contribute to the development of pathology. A key but little tested assumption is that such reactions are metabolically exaggerated. Cardiac activity, using Doppler echocardiography, and oxygen consumption, using mass spectrometry, were measured at rest and during and after a mental stress task and during graded submaximal cycling exercise. Cardiac activity and oxygen consumption showed the expected orderly association during exercise. However, during stress, large increases in cardiac activity were observed in the context of modest rises in energy expenditure; observed cardiac activity during stress substantially exceeded that predicted on the basis of contemporary levels of oxygen consumption. Thus, psychological stress can provoke increases in cardiac activity difficult to account for in terms of the metabolic demands of the stress tas

GLSM's for partial flag manifolds

In this paper we outline some aspects of nonabelian gauged linear sigma models. First, we review how partial flag manifolds (generalizing Grassmannians) are described physically by nonabelian gauged linear sigma models, paying attention to realizations of tangent bundles and other aspects pertinent to (0,2) models. Second, we review constructions of Calabi-Yau complete intersections within such flag manifolds, and properties of the gauged linear sigma models. We discuss a number of examples of nonabelian GLSM's in which the Kahler phases are not birational, and in which at least one phase is realized in some fashion other than as a complete intersection, extending previous work of Hori-Tong. We also review an example of an abelian GLSM exhibiting the same phenomenon. We tentatively identify the mathematical relationship between such non-birational phases, as examples of Kuznetsov's homological projective duality. Finally, we discuss linear sigma model moduli spaces in these gauged linear sigma models. We argue that the moduli spaces being realized physically by these GLSM's are precisely Quot and hyperquot schemes, as one would expect mathematically.Comment: 57 pp, LaTeX; v3: refs added, material on weighted Grassmannians adde

Minimum intrinsic dimension scaling for entropic optimal transport

Motivated by the manifold hypothesis, which states that data with a high extrinsic dimension may yet have a low intrinsic dimension, we develop refined statistical bounds for entropic optimal transport that are sensitive to the intrinsic dimension of the data. Our bounds involve a robust notion of intrinsic dimension, measured at only a single distance scale depending on the regularization parameter, and show that it is only the minimum of these single-scale intrinsic dimensions which governs the rate of convergence. We call this the Minimum Intrinsic Dimension scaling (MID scaling) phenomenon, and establish MID scaling with no assumptions on the data distributions so long as the cost is bounded and Lipschitz, and for various entropic optimal transport quantities beyond just values, with stronger analogs when one distribution is supported on a manifold. Our results significantly advance the theoretical state of the art by showing that MID scaling is a generic phenomenon, and provide the first rigorous interpretation of the statistical effect of entropic regularization as a distance scale.Comment: 53 page