Berezin Kernels and Analysis on Makarevich Spaces

Following ideas of van Dijk and Hille we study the link which exists between maximal degenerate representations and Berezin kernels. We consider the conformal group ${\rm Conf}(V)$ of a simple real Jordan algebra $V$. The maximal degenerate representations $\pi_s$ ($s\in {\mathbb C}$) we shall study are induced by a character of a maximal parabolic subgroup $\bar P$ of ${\rm Conf}(V)$. These representations $\pi_s$ can be realized on a space $I_s$ of smooth functions on $V$. There is an invariant bilinear form ${\mathfrak B}_s$ on the space $I_s$. The problem we consider is to diagonalize this bilinear form ${\mathfrak B}_s$, with respect to the action of a symmetric subgroup $G$ of the conformal group ${\rm Conf}(V)$. This bilinear form can be written as an integral involving the Berezin kernel $B_{\nu}$, an invariant kernel on the Riemannian symmetric space $G/K$, which is a Makarevich symmetric space in the sense of Bertram. Then we can use results by van Dijk and Pevzner who computed the spherical Fourier transform of $B_{\nu}$. From these, one deduces that the Berezin kernel satisfies a remarkable Bernstein identity : $$D(\nu)B_{\nu} =b(\nu)B_{\nu +1},$$ where $D(\nu)$ is an invariant differential operator on $G/K$ and $b(\nu)$ is a polynomial. By using this identity we compute a Hua type integral which gives the normalizing factor for an intertwining operator from $I_{-s}$ to $I_s$. Furthermore we obtain the diagonalization of the invariant bilinear form with respect to the action of the maximal compact group $U$ of the conformal group ${\rm Conf}(V)$

Weighted Bergman kernels and virtual Bergman kernels

We introduce the notion of "virtual Bergman kernel" and apply it to the computation of the Bergman kernel of "domains inflated by Hermitian balls", in particular when the base domain is a bounded symmetric domain.Comment: 12 pages. One-hour lecture for graduate students, SCV 2004, August 2004, Beijing, P.R. China. V2: typo correcte

Balanced metrics on Cartan and Cartan-Hartogs domains

This paper consists of two results dealing with balanced metrics (in S. Donaldson terminology) on nonconpact complex manifolds. In the first one we describe all balanced metrics on Cartan domains. In the second one we show that the only Cartan-Hartogs domain which admits a balanced metric is the complex hyperbolic space. By combining these results with those obtained in [13] (Kaehler-Einstein submanifolds of the infinite dimensional projective space, to appear in Mathematische Annalen) we also provide the first example of complete, Kaehler-Einstein and projectively induced metric g such that $\alpha g$ is not balanced for all $\alpha >0$.Comment: 11 page

A NWB-based dataset and processing pipeline of human single-neuron activity during a declarative memory task

A challenge for data sharing in systems neuroscience is the multitude of different data formats used. Neurodata Without Borders: Neurophysiology 2.0 (NWB:N) has emerged as a standardized data format for the storage of cellular-level data together with meta-data, stimulus information, and behavior. A key next step to facilitate NWB:N adoption is to provide easy to use processing pipelines to import/export data from/to NWB:N. Here, we present a NWB-formatted dataset of 1863 single neurons recorded from the medial temporal lobes of 59 human subjects undergoing intracranial monitoring while they performed a recognition memory task. We provide code to analyze and export/import stimuli, behavior, and electrophysiological recordings to/from NWB in both MATLAB and Python. The data files are NWB:N compliant, which affords interoperability between programming languages and operating systems. This combined data and code release is a case study for how to utilize NWB:N for human single-neuron recordings and enables easy re-use of this hard-to-obtain data for both teaching and research on the mechanisms of human memory

Frequency of Drug Resistance Gene Amplification in Clinical Leishmania Strains

Experimental studies about Leishmania resistance to metal and antifolates have pointed out that gene amplification is one of the main mechanisms of drug detoxification. Amplified genes code for adenosine triphosphate-dependent transporters (multidrug resistance and P-glycoproteins P), enzymes involved in trypanothione pathway, particularly gamma glutamyl cysteine synthase, and others involved in folates metabolism, such as dihydrofolate reductase and pterine reductase. The aim of this study was to detect and quantify the amplification of these genes in clinical strains of visceral leishmaniasis agents: Leishmania infantum, L. donovani, and L. archibaldi. Relative quantification experiments by means of real-time polymerase chain reaction showed that multidrug resistance gene amplification is the more frequent event. For P-glycoproteins P and dihydrofolate reductase genes, level of amplification was comparable to the level observed after in vitro selection of resistant clones. Gene amplification is therefore a common phenomenon in wild strains concurring to Leishmania genomic plasticity. This finding, which corroborates results of experimental studies, supports a better understanding of metal resistance selection and spreading in endemic areas

Metric trees of generalized roundness one

Every finite metric tree has generalized roundness strictly greater than one. On the other hand, some countable metric trees have generalized roundness precisely one. The purpose of this paper is to identify some large classes of countable metric trees that have generalized roundness precisely one. At the outset we consider spherically symmetric trees endowed with the usual combinatorial metric (SSTs). Using a simple geometric argument we show how to determine decent upper bounds on the generalized roundness of finite SSTs that depend only on the downward degree sequence of the tree in question. By considering limits it follows that if the downward degree sequence $(d_{0}, d_{1}, d_{2}...)$ of a SST $(T,\rho)$ satisfies $|\{j \, | \, d_{j} > 1 \}| = \aleph_{0}$, then $(T,\rho)$ has generalized roundness one. Included among the trees that satisfy this condition are all complete $n$-ary trees of depth $\infty$ ($n \geq 2$), all $k$-regular trees ($k \geq 3$) and inductive limits of Cantor trees. The remainder of the paper deals with two classes of countable metric trees of generalized roundness one whose members are not, in general, spherically symmetric. The first such class of trees are merely required to spread out at a sufficient rate (with a restriction on the number of leaves) and the second such class of trees resemble infinite combs.Comment: 14 pages, 2 figures, 2 table

Topological wave functions and heat equations

It is generally known that the holomorphic anomaly equations in topological string theory reflect the quantum mechanical nature of the topological string partition function. We present two new results which make this assertion more precise: (i) we give a new, purely holomorphic version of the holomorphic anomaly equations, clarifying their relation to the heat equation satisfied by the Jacobi theta series; (ii) in cases where the moduli space is a Hermitian symmetric tube domain $G/K$, we show that the general solution of the anomaly equations is a matrix element \IP{\Psi | g | \Omega} of the Schr\"odinger-Weil representation of a Heisenberg extension of $G$, between an arbitrary state $\bra{\Psi}$ and a particular vacuum state $\ket{\Omega}$. Based on these results, we speculate on the existence of a one-parameter generalization of the usual topological amplitude, which in symmetric cases transforms in the smallest unitary representation of the duality group $G'$ in three dimensions, and on its relations to hypermultiplet couplings, nonabelian Donaldson-Thomas theory and black hole degeneracies.Comment: 50 pages; v2: small typos fixed, references added; v3: cosmetic changes, published version; v4: typos fixed, small clarification adde

Temporal Dissociation between Myeloperoxidase (MPO)-Modified LDL and MPO Elevations during Chronic Sleep Restriction and Recovery in Healthy Young Men

OBJECTIVES: Many studies have evaluated the ways in which sleep disturbances may influence inflammation and the possible links of this effect to cardiovascular risk. Our objective was to investigate the effects of chronic sleep restriction and recovery on several blood cardiovascular biomarkers. METHODS AND RESULTS: Nine healthy male non-smokers, aged 22-29 years, were admitted to the Sleep Laboratory for 11 days and nights under continuous electroencephalogram polysomnography. The study consisted of three baseline nights of 8 hours sleep (from 11 pm to 7 am), five sleep-restricted nights, during which sleep was allowed only between 1 am and 6 am, and three recovery nights of 8 hours sleep (11 pm to 7 am). Myeloperoxidase-modified low-density lipoprotein levels increased during the sleep-restricted period indicating an oxidative stress. A significant increase in the quantity of slow-wave sleep was measured during the first recovery night. After this first recovery night, insulin-like growth factor-1 levels increased and myeloperoxidase concentration peaked. CONCLUSIONS: We observed for the first time that sleep restriction and the recovery process are associated with differential changes in blood biomarkers of cardiovascular disease

Maximal surface group representations in isometry groups of classical Hermitian symmetric spaces

Higgs bundles and non-abelian Hodge theory provide holomorphic methods with which to study the moduli spaces of surface group representations in a reductive Lie group G. In this paper we survey the case in which G is the isometry group of a classical Hermitian symmetric space of non-compact type. Using Morse theory on the moduli spaces of Higgs bundles, we compute the number of connected components of the moduli space of representations with maximal Toledo invariant.Comment: v2: added due credits to the work of Burger, Iozzi and Wienhard. v3: corrected count of connected components for G=SU(p,q) (p \neq q); added due credits to the work of Xia and Markman-Xia; minor corrections and clarifications. 31 page

Notes on Stein-Sahi representations and some problems of non L2L^2 harmonic analysis

We discuss one natural class of kernels on pseudo-Riemannian symmetric spaces.Comment: 40p