No Escape from Vardanyan's theorem

Vardanyan's theorem states that the set of PA-valid principles of Quantified Modal Logic, QML, is complete Π⁰₂. We generalize this result to a wide class of theories. The crucial step in the generalization is avoiding the use of Tennenbaum's Theorem.     

Determination of star bodies from p-centroid bodies

In this paper, we prove that an origin-symmetric star body is uniquely determined by its p-centroid body. Furthermore, using spherical harmonics, we establish a result for non-symmetric star bodies. As an application, we show that there is a unique member of $\Gamma_p\langle K\rangle$ characterized by having larger volume than any other member, for all real p?≥?1 that are not even natural numbers, where $\Gamma_p\langle K\rangle$ denotes the p-centroid equivalence class of the star body K.     

Capture of photons and slow uncharged particles by a spherically symmetric charged compact body in the relativistic theory of gravitation

Expressions are obtained in the framework of the relativistic theory of gravitation for the cross sections for capture of photons and slow uncharged particles by a spherically symmetric compact charged body. A comparison with the corresponding relations obtained in the framework of the general theory of relativity is made.Institute of Theoretical and Experimental Physics. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 90, No. 1, pp. 148–154, January, 1992.     

Normal Extensions Escape from the Class of Weighted Shifts on Directed Trees

A formally normal weighted shift on a directed tree is shown to be a bounded normal operator. The question of whether a normal extension of a subnormal weighted shift on a directed tree can be modeled as a weighted shift on some, possibly different, directed tree is answered.     

Reconstruction of convex bodies from surface tensors

We present two algorithms for reconstruction of the shape of convex bodies in the two-dimensional Euclidean space. The first reconstruction algorithm requires knowledge of the exact surface tensors of a convex body up to rank s for some natural number s. When only measurements subject to noise of surface tensors are available for reconstruction, we recommend to use certain values of the surface tensors, namely harmonic intrinsic volumes instead of the surface tensors evaluated at the standard basis. The second algorithm we present is based on harmonic intrinsic volumes and allows for noisy measurements. From a generalized version of Wirtinger's inequality, we derive stability results that are utilized to ensure consistency of both reconstruction procedures. Consistency of the reconstruction procedure based on measurements subject to noise is established under certain assumptions on the noise variables.     

Approximation of convex bodies from chord functions

In this paper we describe a method for computing derivatives of the polar representations of the boundary of a convex body, from generalized chord functions at two points.     

The Safest Escape problem

In this paper, the Safest Escape (SEscape) problem is defined for providing evacuation plans for emergency egress from large buildings or a geographical region. The objective of the SEscape problem is to determine the set of paths and number of evacuees to send along each path such that the minimum probability of arrival at an exit for any evacuee is maximized. Such paths minimize the risk incurred by the evacuees who are forced to take the greatest risk. The problem is considered in a dynamic and time-varying network, where arc capacities are recaptured over time, arc traversal times are time-varying and arc capacities are random variables with probability distribution functions that vary with time. An exact algorithm, the SEscape algorithm, is proposed to address this problem.     

Reconstruction of n-dimensional convex bodies from surface tensors

In this paper, we derive uniqueness and stability results for surface tensors. Further, we develop two algorithms that reconstruct shape of n-dimensional convex bodies. One algorithm requires knowledge of a finite number of surface tensors, whereas the other algorithm is based on noisy measurements of a finite number of harmonic intrinsic volumes. The derived stability results ensure consistency of the two algorithms. Examples that illustrate the feasibility of the algorithms are presented.     

High energy photons from the Crab nebula due to the universal microwave field

The flux of high energy photons emanating from the Crab nebula due to inverse Compton scattering of the universal microwave photons from the high energy electrons in the nebula has been calculated. This flux is compared with experimentally obtained values at certain energies and also with the flux calculated due to Compton-synchrotron process. The flux of gamma-rays of about 100 MeV due to the scattering of the microwave photons is found to be of the same order as that due to the Compton-synchrotron process.     

Escape distribution for an inclined billiard

Hénon [8] used an inclined billiard to investigate aspects of chaotic scattering which occur in satellite encounters and in other situations. His model consisted of a piecewise mapping which described the motion of a point particle bouncing elastically on two disks. A one parameter family of orbits, named h-orbits, was obtained by starting the particle at rest from a given height. We obtain an analytical expression for the escape distribution of the h-orbits, which is also compared with results from numerical simulations. Finally, some discussion is made about possible applications of the h-orbits in connection with Hill’s problem.     

Escape probabilities for slowly recurrent sets

Summary A setAZ ^d (d>-3) is defined to be slowly recurrent for simple random walk if it is recurrent but the probability of enteringA{z:n<|z|<-2n} tends to zero asn. A method is given to estimate escape probabilities for such sets, i.e., the probability of leaving the ball of radiusn without entering the set. The methods are applied to two examples. First, half-lines and finite unions of half-lines inZ ³ are considered. The second example is a random walk path in four dimensions. In the latter case it is proved that the probability that two random walk paths reach the ball of radiusn without intersecting is asymptotic toc(lnn)^–1/2, improving a result of the author.Research partially supported by the National Science Foundation     

The determination of convex bodies from derivatives of section functions

It is known that non-symmetric convex bodies generally cannot be characterized by the volumes of hyperplane sections through one interior point. Falconer and Gardner, however, independently proved that volumes of hyperplane sections through two different interior points determine the body uniquely. We prove that if −1 < q < n − 1 is not an integer, then the derivatives of the order q at zero of parallel section functions at one interior point completely characterize convex bodies in . If 0 ≤ q < n − 1 is an integer then one needs the derivatives of order q at two different interior points (except for the case where q = n − 2, q odd), generalizing the results of Falconer and Gardner. The first named author was partially supported by the NSF grant DMS 0455696. Received: 31 January 2006     

Approximation of convex bodies by axially symmetric bodies

Let be an arbitrary planar convex body. We prove that contains an axially symmetric convex body of area at least . Also approximation by some specific axially symmetric bodies is considered. In particular, we can inscribe a rhombus of area at least in , and we can circumscribe a homothetic rhombus of area at most about . The homothety ratio is at most . Those factors and , as well as the ratio , cannot be improved.

     

An algorithm for reconstructing convex bodies from their projections

An algorithm is described for the approximative reconstruction of a plane convex body from its projections in a finite number of directions.A priori anda posteriori error estimates are given, and the convergence of certain sequences of an approximative solution of the reconstruction problem to the exact solution is proven. Finally, it is shown that, after small modifications, the algorithm can be applied to reconstruct convex bodies from discrete projectional data.The algorithm consists in an approximation of the convex body, to be reconstructed, by recursively defined cores and envelopes, following the ideas of Kuba [6] for the reconstruction of binary patterns.This paper was started at the University of Erlangen-Nürnberg in 1983–1984, when the second author was a fellow of the Alexander von Humboldt Foundation, while the last author was supported by the German Research Council (DFG, Contract Ko-506/8-1).