Abstract:
We give arguments for a conjecture made in a previous paper, that one has to use only the gauged sugra action for the calculation of correlators of certain operators via the AdS-CFT correspondence. The existence of consistent truncations implies that the massive modes decouple, and gauged supergravity is sufficient for computing n-point functions of CFT operators coupled to the massless (sugra) sector. The action obtained from the linear ansatz, of the type $\phi(x,y)=\phi_I(x)Y^I(y)$ gives only part of the gauged sugra. This means that there is a difference for the correlators on the boundary of AdS space. We find, studying examples of correlators, that the right prescription is to use the full gauged sugra, which implies using the full nonlinear KK ansatz. To this purpose, we analyze 3 point functions of various gauge fields in 5 and 7 dimensions, and the R-current anomaly in the corresponding CFT. We also show that the nonlinear rotation in the tower of scalar fields of Lee et al., Corrado et al. and Bastianelli and Zucchini produces a consistent truncation to the massless level and coincides with the Taylor expansion of the nonlinear KK ansatz in massless scalar fluctuations. Finally, we speculate about the way to do the full nonlinear rotation for the massive tower.

Abstract:
We present exact solutions to the Einstein-Maxwell system of equations in spherically symmetric gravitational fields with a specified form of the electric field intensity. The condition of pressure isotropy yields a difference equation with variable, rational coefficients. In an earlier treatment this condition was integrated by first transforming it to a hypergeometric equation. We demonstrate that it is possible to obtain a more general class of solutions to the Einstein-Maxwell system both in the form of special functions and elementary functions. Our results contain particular solutions found previously including models of charged relativistic spheres and uncharged neutron star models.

Abstract:
We study a novel five-dimensional, {\it N}=5 supergravity in the context of Lie superalgebra SU(5/2). The possible successive superalgebraic truncations from {\it N}=5 theory to the lower supersymmetric {\it N}=4,3,2, and 1 supergravity theories are systematically analyzed as a sub-superalgebraic chain of SU(5/2)$\supset$ SU(4/2) $\supset$ SU(3/2) $\supset$ SU(2/2) $\supset$ SU(1/2) by using the Kac-Dynkin weight techniques.

Abstract:
Consider a set of discounted optimal stopping problems for a one-parameter family of objective functions and a fixed diffusion process, started at a fixed point. A standard problem in stochastic control/optimal stopping is to solve for the problem value in this setting. In this article we consider an inverse problem; given the set of problem values for a family of objective functions, we aim to recover the diffusion. Under a natural assumption on the family of objective functions we can characterise existence and uniqueness of a diffusion for which the optimal stopping problems have the specified values. The solution of the problem relies on techniques from generalised convexity theory

Abstract:
The presence of radial truncations in stellar disks is reviewed. There is ample evidence that many disk galaxies have relatively shaprt truncations in their disks. These often are symmetric and independent of the wavelength band of the observations. The ratio of the truncation radius R_{max} to the disk scalelength h appears often less then 4.5, as expected on a simple model for the disk collapse. Current samples of galaxies observed may however not be representative and heavily biased towards disks with the largest scalelengths. Many spiral galaxies also have HI warps and these generally start at the truncation radius of the stellar disk. The HI surface density suddenly becomes much flatter with radius. In some galaxies the start of the warp and the position of the disk truncation radius is accompanied by a drop in the rotation velocity. In the regiosn beyond the dis truncation in the HI layer some star formation does occur, but the heavy element abundance and the dust content are very low. All evidence is consistent with the notion that the outer gas parts of the disks constitute recently accreted material, at least accreted after formation of what is now the stellar thin disk. Although various models exist for the origin of the truncations in the stellar disks, this at present remains unclear.

Abstract:
It is known that positive definiteness is not enough for the multidimensional moment problem to be solved. We would like throw in to the garden of existing in this matter so far results one more, a result which takes into considerations the utmost possible truncations.

Abstract:
We extend to multilinear Hankel operators the fact that some truncations of bounded Hankel operators are bounded. We prove and use a continuity property of bilinear Hilbert transforms on products of Lipschitz spaces and Hardy spaces.

Abstract:
Statistical properties of non--symmetric real random matrices of size $M$, obtained as truncations of random orthogonal $N\times N$ matrices are investigated. We derive an exact formula for the density of eigenvalues which consists of two components: finite fraction of eigenvalues are real, while the remaining part of the spectrum is located inside the unit disk symmetrically with respect to the real axis. In the case of strong non--orthogonality, $M/N=$const, the behavior typical to real Ginibre ensemble is found. In the case $M=N-L$ with fixed $L$, a universal distribution of resonance widths is recovered.

Abstract:
We associate to each $r$-multigraded, locally finitely generated ideal in the "large polynomial ring" on countably many indeterminates a power series in $r$ variables; this power series is the limit in the adic topology of the numerators of the rational functions which give the Hilbert series of the truncations of the ideal. We characterise the set of all power series so obtained. Our main technical tools are an approximation result which asserts that truncation and the forming of initial ideals commute in a filtered sense, and standard inclusion/exclusion, M\"obius inversion, and LCM-lattice homology methods generalised to monomial ideals in countably many variables.

Abstract:
We investigate the Exact Renormalization Group (ERG) description of ($Z_2$ invariant) one-component scalar field theory, in the approximation in which all momentum dependence is discarded in the effective vertices. In this context we show how one can perform a systematic search for non-perturbative continuum limits without making any assumption about the form of the lagrangian. Concentrating on the non-perturbative three dimensional Wilson fixed point, we then show that the sequence of truncations $n=2,3,\dots$, obtained by expanding about the field $\varphi=0$ and discarding all powers $\varphi^{2n+2}$ and higher, yields solutions that at first converge to the answer obtained without truncation, but then cease to further converge beyond a certain point. No completely reliable method exists to reject the many spurious solutions that are also found. These properties are explained in terms of the analytic behaviour of the untruncated solutions -- which we describe in some detail.