989 research outputs found
The averaged characteristic polynomial for the Gaussian and chiral Gaussian ensembles with a source
In classical random matrix theory the Gaussian and chiral Gaussian random
matrix models with a source are realized as shifted mean Gaussian, and chiral
Gaussian, random matrices with real , complex ( and
real quaternion ) elements. We use the Dyson Brownian motion model
to give a meaning for general . In the Gaussian case a further
construction valid for is given, as the eigenvalue PDF of a
recursively defined random matrix ensemble. In the case of real or complex
elements, a combinatorial argument is used to compute the averaged
characteristic polynomial. The resulting functional forms are shown to be a
special cases of duality formulas due to Desrosiers. New derivations of the
general case of Desrosiers' dualities are given. A soft edge scaling limit of
the averaged characteristic polynomial is identified, and an explicit
evaluation in terms of so-called incomplete Airy functions is obtained.Comment: 21 page
Explicit formulas for the generalized Hermite polynomials in superspace
We provide explicit formulas for the orthogonal eigenfunctions of the
supersymmetric extension of the rational Calogero-Moser-Sutherland model with
harmonic confinement, i.e., the generalized Hermite (or Hi-Jack) polynomials in
superspace. The construction relies on the triangular action of the Hamiltonian
on the supermonomial basis. This translates into determinantal expressions for
the Hamiltonian's eigenfunctions.Comment: 19 pages. This is a recasting of the second part of the first version
of hep-th/0305038 which has been splitted in two articles. In this revised
version, the introduction has been rewritten and a new appendix has been
added. To appear in JP
Macdonald polynomials in superspace: conjectural definition and positivity conjectures
We introduce a conjectural construction for an extension to superspace of the
Macdonald polynomials. The construction, which depends on certain orthogonality
and triangularity relations, is tested for high degrees. We conjecture a simple
form for the norm of the Macdonald polynomials in superspace, and a rather
non-trivial expression for their evaluation. We study the limiting cases q=0
and q=\infty, which lead to two families of Hall-Littlewood polynomials in
superspace. We also find that the Macdonald polynomials in superspace evaluated
at q=t=0 or q=t=\infty seem to generalize naturally the Schur functions. In
particular, their expansion coefficients in the corresponding Hall-Littlewood
bases appear to be polynomials in t with nonnegative integer coefficients. More
strikingly, we formulate a generalization of the Macdonald positivity
conjecture to superspace: the expansion coefficients of the Macdonald
superpolynomials expanded into a modified version of the Schur superpolynomial
basis (the q=t=0 family) are polynomials in q and t with nonnegative integer
coefficients.Comment: 18 page
Jack superpolynomials with negative fractional parameter: clustering properties and super-Virasoro ideals
The Jack polynomials P_\lambda^{(\alpha)} at \alpha=-(k+1)/(r-1) indexed by
certain (k,r,N)-admissible partitions are known to span an ideal I^{(k,r)}_N of
the space of symmetric functions in N variables. The ideal I^{(k,r)}_N is
invariant under the action of certain differential operators which include half
the Virasoro algebra. Moreover, the Jack polynomials in I^{(k,r)}_N admit
clusters of size at most k: they vanish when k+1 of their variables are
identified, and they do not vanish when only k of them are identified. We
generalize most of these properties to superspace using orthogonal
eigenfunctions of the supersymmetric extension of the trigonometric
Calogero-Moser-Sutherland model known as Jack superpolynomials. In particular,
we show that the Jack superpolynomials P_{\Lambda}^{(\alpha)} at
\alpha=-(k+1)/(r-1) indexed by certain (k,r,N)-admissible superpartitions span
an ideal {\mathcal I}^{(k,r)}_N of the space of symmetric polynomials in N
commuting variables and N anticommuting variables. We prove that the ideal
{\mathcal I}^{(k,r)}_N is stable with respect to the action of the
negative-half of the super-Virasoro algebra. In addition, we show that the Jack
superpolynomials in {\mathcal I}^{(k,r)}_N vanish when k+1 of their commuting
variables are equal, and conjecture that they do not vanish when only k of them
are identified. This allows us to conclude that the standard Jack polynomials
with prescribed symmetry should satisfy similar clustering properties. Finally,
we conjecture that the elements of {\mathcal I}^{(k,2)}_N provide a basis for
the subspace of symmetric superpolynomials in N variables that vanish when k+1
commuting variables are set equal to each other.Comment: 36 pages; the main changes in v2 are : 1) in the introduction, we
present exceptions to an often made statement concerning the clustering
property of the ordinary Jack polynomials for (k,r,N)-admissible partitions
(see Footnote 2); 2) Conjecture 14 is substantiated with the extensive
computational evidence presented in the new appendix C; 3) the various tests
supporting Conjecture 16 are reporte
Supersymmetric Many-particle Quantum Systems with Inverse-square Interactions
The development in the study of supersymmetric many-particle quantum systems
with inverse-square interactions is reviewed. The main emphasis is on quantum
systems with dynamical OSp(2|2) supersymmetry. Several results related to
exactly solved supersymmetric rational Calogero model, including shape
invariance, equivalence to a system of free superoscillators and non-uniqueness
in the construction of the Hamiltonian, are presented in some detail. This
review also includes a formulation of pseudo-hermitian supersymmetric quantum
systems with a special emphasis on rational Calogero model. There are quite a
few number of many-particle quantum systems with inverse-square interactions
which are not exactly solved for a complete set of states in spite of the
construction of infinitely many exact eigen functions and eigenvalues. The
Calogero-Marchioro model with dynamical SU(1,1|2) supersymmetry and a quantum
system related to short-range Dyson model belong to this class and certain
aspects of these models are reviewed. Several other related and important
developments are briefly summarized.Comment: LateX, 65 pages, Added Acknowledgment, Discussions and References,
Version to appear in Jouranl of Physics A: Mathematical and Theoretical
(Commissioned Topical Review Article
Some properties of angular integrals
We find new representations for Itzykson-Zuber like angular integrals for
arbitrary beta, in particular for the orthogonal group O(n), the unitary group
U(n) and the symplectic group Sp(2n). We rewrite the Haar measure integral, as
a flat Lebesge measure integral, and we deduce some recursion formula on n. The
same methods gives also the Shatashvili's type moments. Finally we prove that,
in agreement with Brezin and Hikami's observation, the angular integrals are
linear combinations of exponentials whose coefficients are polynomials in the
reduced variables (x_i-x_j)(y_i-y_j).Comment: 43 pages, Late
Non-intersecting squared Bessel paths: critical time and double scaling limit
We consider the double scaling limit for a model of non-intersecting
squared Bessel processes in the confluent case: all paths start at time
at the same positive value , remain positive, and are conditioned to end
at time at . After appropriate rescaling, the paths fill a region in
the --plane as that intersects the hard edge at at a
critical time . In a previous paper (arXiv:0712.1333), the scaling
limits for the positions of the paths at time were shown to be
the usual scaling limits from random matrix theory. Here, we describe the limit
as of the correlation kernel at critical time and in the
double scaling regime. We derive an integral representation for the limit
kernel which bears some connections with the Pearcey kernel. The analysis is
based on the study of a matrix valued Riemann-Hilbert problem by
the Deift-Zhou steepest descent method. The main ingredient is the construction
of a local parametrix at the origin, out of the solutions of a particular
third-order linear differential equation, and its matching with a global
parametrix.Comment: 53 pages, 15 figure
Two-dimensional superstrings and the supersymmetric matrix model
We present evidence that the supersymmetric matrix model of Marinari and
Parisi represents the world-line theory of N unstable D-particles in type II
superstring theory in two dimensions. This identification suggests that the
matrix model gives a holographic description of superstrings in a
two-dimensional black hole geometry.Comment: 22 pages, 2 figures; v2: corrected eqn 4.6; v3: corrected appendices
and discussion of vacua, added ref
Microvascular response to transfusion in elective spine surgery
AIM: To investigate the microvascular (skeletal muscle tissue oxygenation; SmO2) response to transfusion in patients undergoing elective complex spine surgery.
METHODS: After IRB approval and written informed consent, 20 patients aged 18 to 85 years of age undergoing \u3e 3 level anterior and posterior spine fusion surgery were enrolled in the study. Patients were followed throughout the operative procedure, and for 12 h postoperatively. In addition to standard American Society of Anesthesiologists monitors, invasive measurements including central venous pressure, continual analysis of stroke volume (SV), cardiac output (CO), cardiac index (CI), and stroke volume variability (SVV) was performed. To measure skeletal muscle oxygen saturation (SmO2) during the study period, a non-invasive adhesive skin sensor based on Near Infrared Spectroscopy was placed over the deltoid muscle for continuous recording of optical spectra. All administration of fluids and blood products followed standard procedures at the Hospital for Special Surgery, without deviation from usual standards of care at the discretion of the Attending Anesthesiologist based on individual patient comorbidities, hemodynamic status, and laboratory data. Time stamps were collected for administration of colloids and blood products, to allow for analysis of SmO2 immediately before, during, and after administration of these fluids, and to allow for analysis of hemodynamic data around the same time points. Hemodynamic and oxygenation variables were collected continuously throughout the surgery, including heart rate, blood pressure, mean arterial pressure, SV, CO, CI, SVV, and SmO2. Bivariate analyses were conducted to examine the potential associations between the outcome of interest, SmO2, and each hemodynamic parameter measured using Pearson\u27s correlation coefficient, both for the overall cohort and within-patients individually. The association between receipt of packed red blood cells and SmO2 was performed by running an interrupted time series model, with SmO2 as our outcome, controlling for the amount of time spent in surgery before and after receipt of PRBC and for the inherent correlation between observations. Our model was fit using PROC AUTOREG in SAS version 9.2. All other analyses were also conducted in SAS version 9.2 (SAS Institute Inc., Cary, NC, United States).
RESULTS: Pearson correlation coefficients varied widely between SmO2 and each hemodynamic parameter examined. The strongest positive correlations existed between ScvO2 (P = 0.41) and SV (P = 0.31) and SmO2; the strongest negative correlations were seen between albumin (P = -0.43) and cell saver (P = -0.37) and SmO2. Correlations for other laboratory parameters studied were weak and only based on a few observations. In the final model we found a small, but significant increase in SmO2 at the time of PRBC administration by 1.29 units (P = 0.0002). SmO2 values did not change over time prior to PRBC administration (P = 0.6658) but following PRBC administration, SmO2 values declined significantly by 0.015 units (P \u3c 0.0001).
CONCLUSION: Intra-operative measurement of SmO2 during large volume, yet controlled hemorrhage, does not show a statistically significant correlation with either invasive hemodynamic, or laboratory parameters in patients undergoing elective complex spine surgery
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