Rates of convergence for empirical spectral measures: a soft approach

Understanding the limiting behavior of eigenvalues of random matrices is the central problem of random matrix theory. Classical limit results are known for many models, and there has been significant recent progress in obtaining more quantitative, non-asymptotic results. In this paper, we describe a systematic approach to bounding rates of convergence and proving tail inequalities for the empirical spectral measures of a wide variety of random matrix ensembles. We illustrate the approach by proving asymptotically almost sure rates of convergence of the empirical spectral measure in the following ensembles: Wigner matrices, Wishart matrices, Haar-distributed matrices from the compact classical groups, powers of Haar matrices, randomized sums and random compressions of Hermitian matrices, a random matrix model for the Hamiltonians of quantum spin glasses, and finally the complex Ginibre ensemble. Many of the results appeared previously and are being collected and described here as illustrations of the general method; however, some details (particularly in the Wigner and Wishart cases) are new. Our approach makes use of techniques from probability in Banach spaces, in particular concentration of measure and bounds for suprema of stochastic processes, in combination with more classical tools from matrix analysis, approximation theory, and Fourier analysis. It is highly flexible, as evidenced by the broad list of examples. It is moreover based largely on "soft" methods, and involves little hard analysis

Asymptotics of the mean-field Heisenberg model

We consider the mean-field classical Heisenberg model and obtain detailed information about the total spin of the system by studying the model on a complete graph and sending the number of vertices to infinity. In particular, we obtain Cramer- and Sanov-type large deviations principles for the total spin and the empirical spin distribution and demonstrate a second-order phase transition in the Gibbs measures. We also study the asymptotics of the total spin throughout the phase transition using Stein's method, proving central limit theorems in the sub- and supercritical phases and a nonnormal limit theorem at the critical temperature.Comment: 44 page

Augmenting Exposure Therapy for Social Anxiety with tDCS

Purpose/Background: Exposure therapy is one of the most potent techniques available to treat social anxiety. However, studies suggest that exposure therapy only produces full remission in 20-50% of patients. Furthermore, laboratory conditioning and extinction studies suggest that fear responses toward individuals who differ from one\u27s own ethnicity/race may be more resistant to extinction. Because activation of the medial prefrontal cortex has been associated with facilitating fear reduction during exposure therapy, we expect that targeting activation of this region with a stimulation technique called transcranial direct current stimulation (tDCS) may improve outcomes from exposure therapy for social anxiety. The present study will therefore test the hypotheses that (1) fear responding at baseline will be greater toward an audience that does not match (vs matches) the participant\u27s own ethnicity, (2) pairing exposure therapy with active (vs sham) tDCS will facilitate alleviation of social anxiety symptoms, and (3) pairing exposure therapy with active (vs sham) tDCS facilitates extinction of fear response toward individuals who differ from the participant\u27s own ethnicity. Materials & Methods: We are recruiting Latino and non-Latino/Caucasian undergraduates with a fear of public speaking, the most commonly feared situation among individuals with social anxiety. Participants (N = 128) will receive either active/anodal (n = 64) or sham (n = 64) tDCS stimulation targeting the mPFC during an exposure therapy session delivered through virtual reality (VR). During exposure therapy, participants will complete six, 3-minute public speaking trials, alternating in a randomized order between audiences that are 75% matched to the participant\u27s ethnicity and 75% unmatched to the participant\u27s ethnicity. At one-month follow up, participants will complete two behavioral avoidance tests (BATs) parallel to therapy procedures, with one ethnic-matched trial and one ethnic-unmatched trial. Fear response during each BAT will be assessed behaviorally (duration of speech), physiologically (heart rate variability and electrodermal response), and subjectively (peak fear rating, on a 0 to 100 scale). At baseline and one-month follow-up, participants will also complete a battery of social anxiety questionnaires. Results: We will present methods and preliminary findings from the study. Results will include a preliminary examination of whether fear responding is greater toward individuals who differ from (vs match) the participant\u27s own ethnicity, whether pairing exposure therapy with active (vs sham) tDCS facilitates alleviation of social anxiety symptoms overall, and whether pairing exposure therapy with active (vs sham) tDCS facilitates alleviation of social anxiety responding toward individuals who differ from (vs match) the participant\u27s own ethnicity. Discussion/Conclusion: Findings point to key strategies to improve outcomes from exposure therapy for social anxiety, and could also have implications for improving response to exposure-based therapies for other anxiety disorders. Furthermore, if tDCS facilitates reductions in fear response toward ethnic/racial out-groups, minority/Latino individuals may experience better generalization of treatment effects for daily-life scenarios (in which they are surrounded by outgroup members), and ethnic/racial majority individuals will be better able to contribute to an inclusive social environment

Rate of convergence of linear functions on the unitary group

We study the rate of convergence to a normal random variable of the real and imaginary parts of Tr(AU), where U is an N x N random unitary matrix and A is a deterministic complex matrix. We show that the rate of convergence is O(N^{-2 + b}), with 0 <= b < 1, depending only on the asymptotic behaviour of the singular values of A; for example, if the singular values are non-degenerate, different from zero and O(1) as N -> infinity, then b=0. The proof uses a Berry-Esse'en inequality for linear combinations of eigenvalues of random unitary, matrices, and so appropriate for strongly dependent random variables.Comment: 34 pages, 1 figure; corrected typos, added remark 3.3, added 3 reference

Oxygen Cost of Performing Selected Adult and Child Care Activities

International Journal of Exercise Science 6(1) : 11-19, 2013. Little is known about the oxygen cost of caring for infants and older adults. Many people perform these activities so it is useful to know the energy cost and if the activities are of sufficient intensity to contribute to meeting physical activity recommendations. The purpose of this study was to assess the oxygen cost of four care-related activities in the Compendium of Physical Activities. Nineteen participants (n = 10 women, n = 9 men; Age = 36.4 ± 13.6 y; % Fat = 34.1 ± 10.5; BMI = 28.1 ± 4.5 kg/m2) performed four activities: 1) pushing an infant in a stroller, 2) pushing an adult in a wheelchair, 3) carrying an infant, and 4) bathing and dressing an infant. The oxygen cost was assessed using a portable metabolic unit. Activities were performed in random order for 8 minutes. The oxygen cost and heart rates, respectively, for healthy adults during care related activities were 3.09 METs and 90 ± 8 beats per minute (bpm) for pushing an infant in a stroller, 3.69 METs and 97 ± 9 bpm for pushing an adult in a wheelchair, 2.37 METs and 85 ± 9 bpm for carrying an infant, and 2.00 METs and 87 ± 9 bpm for bathing and dressing an infant. Carrying an infant and bathing an infant are light-intensity physical activities and pushing a wheelchair or a stroller are moderate intensity activities. The latter activities are of sufficient intensity to meet health-related physical activity recommendations

Central limit theorem for multiplicative class functions on the symmetric group

Hambly, Keevash, O'Connell and Stark have proven a central limit theorem for the characteristic polynomial of a permutation matrix with respect to the uniform measure on the symmetric group. We generalize this result in several ways. We prove here a central limit theorem for multiplicative class functions on symmetric group with respect to the Ewens measure and compute the covariance of the real and the imaginary part in the limit. We also estimate the rate of convergence with the Wasserstein distance.Comment: 23 pages; the mathematics is the same as in the previous version, but there are several improvments in the presentation, including a more intuitve name for the considered function

Stein's Method and Characters of Compact Lie Groups

Stein's method is used to study the trace of a random element from a compact Lie group or symmetric space. Central limit theorems are proved using very little information: character values on a single element and the decomposition of the square of the trace into irreducible components. This is illustrated for Lie groups of classical type and Dyson's circular ensembles. The approach in this paper will be useful for the study of higher dimensional characters, where normal approximations need not hold.Comment: 22 pages; same results, but more efficient exposition in Section 3.

On Poincare and logarithmic Sobolev inequalities for a class of singular Gibbs measures

This note, mostly expository, is devoted to Poincar{\'e} and log-Sobolev inequalities for a class of Boltzmann-Gibbs measures with singular interaction. Such measures allow to model one-dimensional particles with confinement and singular pair interaction. The functional inequalities come from convexity. We prove and characterize optimality in the case of quadratic confinement via a factorization of the measure. This optimality phenomenon holds for all beta Hermite ensembles including the Gaussian unitary ensemble, a famous exactly solvable model of random matrix theory. We further explore exact solvability by reviewing the relation to Dyson-Ornstein-Uhlenbeck diffusion dynamics admitting the Hermite-Lassalle orthogonal polynomials as a complete set of eigenfunctions. We also discuss the consequence of the log-Sobolev inequality in terms of concentration of measure for Lipschitz functions such as maxima and linear statistics.Comment: Minor improvements. To appear in Geometric Aspects of Functional Analysis -- Israel Seminar (GAFA) 2017-2019", Lecture Notes in Mathematics 225

Microparticle-mediated transfer of the viral receptors CAR and CD46, and the CFTR channel in a CHO cell model confers new functions to target cells

Cell microparticles (MPs) released in the extracellular milieu can embark plasma membrane and intracellular components which are specific of their cellular origin, and transfer them to target cells. The MP-mediated, cell-to-cell transfer of three human membrane glycoproteins of different degrees of complexity was investigated in the present study, using a CHO cell model system. We first tested the delivery of CAR and CD46, two monospanins which act as adenovirus receptors, to target CHO cells. CHO cells lack CAR and CD46, high affinity receptors for human adenovirus serotype 5 (HAdV5), and serotype 35 (HAdV35), respectively. We found that MPs derived from CHO cells (MP-donor cells) constitutively expressing CAR (MP-CAR) or CD46 (MP-CD46) were able to transfer CAR and CD46 to target CHO cells, and conferred selective permissiveness to HAdV5 and HAdV35. In addition, target CHO cells incubated with MP-CD46 acquired the CD46-associated function in complement regulation. We also explored the MP-mediated delivery of a dodecaspanin membrane glycoprotein, the CFTR to target CHO cells. CFTR functions as a chloride channel in human cells and is implicated in the genetic disease cystic fibrosis. Target CHO cells incubated with MPs produced by CHO cells constitutively expressing GFP-tagged CFTR (MP-GFP-CFTR) were found to gain a new cellular function, the chloride channel activity associated to CFTR. Time-course analysis of the appearance of GFP-CFTR in target cells suggested that MPs could achieve the delivery of CFTR to target cells via two mechanisms: the transfer of mature, membrane-inserted CFTR glycoprotein, and the transfer of CFTR-encoding mRNA. These results confirmed that cell-derived MPs represent a new class of promising therapeutic vehicles for the delivery of bioactive macromolecules, proteins or mRNAs, the latter exerting the desired therapeutic effect in target cells via de novo synthesis of their encoded proteins