83 research outputs found
Covariant q-differential operators and unitary highest weight representations for U_q su(n,n)
We investigate a one-parameter family of quantum Harish-Chandra modules of
U_q sl(2n). This family is an analog of the holomorphic discrete series of
representations of the group SU(n,n) for the quantum group U_q su(n, n). We
introduce a q-analog of "the wave" operator (a determinant-type differential
operator) and prove certain covariance property of its powers. This result is
applied to the study of some quotients of the above-mentioned quantum
Harish-Chandra modules. We also prove an analog of a known result by J.Faraut
and A.Koranyi on the expansion of reproducing kernels which determines the
analytic continuation of the holomorphic discrete series.Comment: 26 page
Group Theory and Quasiprobability Integrals of Wigner Functions
The integral of the Wigner function of a quantum mechanical system over a
region or its boundary in the classical phase plane, is called a
quasiprobability integral. Unlike a true probability integral, its value may
lie outside the interval [0,1]. It is characterized by a corresponding
selfadjoint operator, to be called a region or contour operator as appropriate,
which is determined by the characteristic function of that region or contour.
The spectral problem is studied for commuting families of region and contour
operators associated with concentric disks and circles of given radius a. Their
respective eigenvalues are determined as functions of a, in terms of the
Gauss-Laguerre polynomials. These polynomials provide a basis of vectors in
Hilbert space carrying the positive discrete series representations of the
algebra su(1,1)or so(2,1). The explicit relation between the spectra of
operators associated with disks and circles with proportional radii, is given
in terms of the dicrete variable Meixner polynomials.Comment: 11 pages, latex fil
Quantum planes and quantum cylinders from Poisson homogeneous spaces
Quantum planes and a new quantum cylinder are obtained as quantization of
Poisson homogeneous spaces of two different Poisson structures on classical
Euclidean group E(2).Comment: 13 pages, plain Tex, no figure
On a correspondence between quantum SU(2), quantum E(2) and extended quantum SU(1,1)
In a previous paper, we showed how one can obtain from the action of a
locally compact quantum group on a type I-factor a possibly new locally compact
quantum group. In another paper, we applied this construction method to the
action of quantum SU(2) on the standard Podles sphere to obtain Woronowicz'
quantum E(2). In this paper, we will apply this technique to the action of
quantum SU(2) on the quantum projective plane (whose associated von Neumann
algebra is indeed a type I-factor). The locally compact quantum group which
then comes out at the other side turns out to be the extended SU(1,1) quantum
group, as constructed by Koelink and Kustermans. We also show that there exists
a (non-trivial) quantum groupoid which has at its corners (the duals of) the
three quantum groups mentioned above.Comment: 35 page
Free Meixner states
Free Meixner states are a class of functionals on non-commutative polynomials
introduced in math.CO/0410482. They are characterized by a resolvent-type form
for the generating function of their orthogonal polynomials, by a recursion
relation for those polynomials, or by a second-order non-commutative
differential equation satisfied by their free cumulant functional. In this
paper, we construct an operator model for free Meixner states. By combinatorial
methods, we also derive an operator model for their free cumulant functionals.
This, in turn, allows us to construct a number of examples. Many of these
examples are shown to be trivial, in the sense of being free products of
functionals which depend on only a single variable, or rotations of such free
products. On the other hand, the multinomial distribution is a free Meixner
state and is not a product. Neither is a large class of tracial free Meixner
states which are analogous to the simple quadratic exponential families in
statistics.Comment: 30 page
Spherical Functions Associated With the Three Dimensional Sphere
In this paper, we determine all irreducible spherical functions \Phi of any K
-type associated to the pair (G,K)=(\SO(4),\SO(3)). This is accomplished by
associating to \Phi a vector valued function H=H(u) of a real variable u, which
is analytic at u=0 and whose components are solutions of two coupled systems of
ordinary differential equations. By an appropriate conjugation involving Hahn
polynomials we uncouple one of the systems. Then this is taken to an uncoupled
system of hypergeometric equations, leading to a vector valued solution P=P(u)
whose entries are Gegenbauer's polynomials. Afterward, we identify those
simultaneous solutions and use the representation theory of \SO(4) to
characterize all irreducible spherical functions. The functions P=P(u)
corresponding to the irreducible spherical functions of a fixed K-type \pi_\ell
are appropriately packaged into a sequence of matrix valued polynomials
(P_w)_{w\ge0} of size (\ell+1)\times(\ell+1). Finally we proved that \widetilde
P_w={P_0}^{-1}P_w is a sequence of matrix orthogonal polynomials with respect
to a weight matrix W. Moreover we showed that W admits a second order symmetric
hypergeometric operator \widetilde D and a first order symmetric differential
operator \widetilde E.Comment: 49 pages, 2 figure
Askey-Wilson Type Functions, With Bound States
The two linearly independent solutions of the three-term recurrence relation
of the associated Askey-Wilson polynomials, found by Ismail and Rahman in [22],
are slightly modified so as to make it transparent that these functions satisfy
a beautiful symmetry property. It essentially means that the geometric and the
spectral parameters are interchangeable in these functions. We call the
resulting functions the Askey-Wilson functions. Then, we show that by adding
bound states (with arbitrary weights) at specific points outside of the
continuous spectrum of some instances of the Askey-Wilson difference operator,
we can generate functions that satisfy a doubly infinite three-term recursion
relation and are also eigenfunctions of -difference operators of arbitrary
orders. Our result provides a discrete analogue of the solutions of the purely
differential version of the bispectral problem that were discovered in the
pioneering work [8] of Duistermaat and Gr\"unbaum.Comment: 42 pages, Section 3 moved to the end, minor correction
On the Two q-Analogue Logarithmic Functions
There is a simple, multi-sheet Riemann surface associated with e_q(z)'s
inverse function ln_q(w) for 0< q < 1. A principal sheet for ln_q(w) can be
defined. However, the topology of the Riemann surface for ln_q(w) changes each
time "q" increases above the collision point of a pair of the turning points of
e_q(x). There is also a power series representation for ln_q(1+w). An
infinite-product representation for e_q(z) is used to obtain the ordinary
natural logarithm ln{e_q(z)} and the values of sum rules for the zeros "z_i" of
e_q(z). For |z|<|z_1|, e_q(z)=exp{b(z)} where b(z) is a simple, explicit power
series in terms of values of these sum rules. The values of the sum rules for
the q-trigonometric functions, sin_q(z) and cos_q(z), are q-deformations of the
usual Bernoulli numbers.Comment: This is the final version to appear in J.Phys.A: Math. & General.
Some explict formulas added, and to update the reference
Clinical significance of stromal apoptosis in colorectal cancer
BackgroundEpithelial and stromal cells play an important role in the development of colorectal cancer (CRC). We aimed to determine the prognostic significance of both epithelial and stromal cell apoptosis in CRC.MethodsTotal apoptosis was determined by caspase-3 activity measurements in protein homogenates of CRC specimens and adjacent normal mucosa of 211 CRC patients. Epithelial apoptosis was determined by an ELISA specific for a caspase-3-degraded cytokeratin 18 product, the M30 antigen. Stromal apoptosis was determined from the ratio between total and epithelial apoptosis.ResultsEpithelial and stromal apoptosis, as well as total apoptosis, were significantly higher in CRC compared with corresponding adjacent normal mucosa. Low total tumour apoptosis (< or = median caspase-3 activity) was associated with a significantly worse disease recurrence (hazard ratio (HR), 95% confidence interval (95% CI): 1.77 (1.05-3.01)), independent of clinocopathological parameters. Epithelial apoptosis was not associated with clinical outcome. In contrast, low stromal apoptosis (< or = median caspase-3/M30) was found to be an independent prognostic factor for overall survival, disease-free survival and disease recurrence, with HRs (95% CI) of 1.66 (1.17-2.35), 1.62 (1.15-2.29) and 1.69 (1.01-2.85), respectively.InterpretationStromal apoptosis, in contrast to epithelial apoptosis, is an important factor with respect to survival and disease-recurrence in CRC
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