The Generalized Radon Transform on the Plane, the Inverse Transform, and the Cavalieri Conditions

In the two-dimensional case, the generalized Radon transform takes each function supported in a disk to the values of the integrals of that function over a family of curves. We assume that the curves differ only slightly from straight lines and the network formed by these curves has the same topological structure as the network of straight lines. Thus, the generalized Radon transform specifies a function on the set of straight lines. Under these conditions, we obtain a solution of the inversion problem for the generalized Radon transform and indicate a Cavalieri condition describing the range of this transform in the space of functions on the set of straight lines.     

Inverse Backscattering Problem for the Acoustic Equation in Even Dimensions

We show that the sound speedc(x) of the acoustic wave equation in any even dimension can be uniquely determined by the backscattering data provided that it is close to a constant. In the three-dimensional case, P. Stefanov and G. Uhlmann (SIAM J. Math. Anal.28,1997, 1191–1204) have proved a similar result. Their method takes advantage of the inversion formula for the Radon transform in odd dimensions being a local operator. This is not true in even dimensions. Moreover, the odd-dimensional Lax and Phillips modified Radon transform fails to work in even dimensions. In this paper, we overcome these difficulties and prove an even-dimensional version of Stefanov and Uhlmann's result.     

Moment matching approximation of Asian basket option prices

In this paper we propose some moment matching pricing methods for European-style discrete arithmetic Asian basket options in a Black & Scholes framework. We generalize the approach of [M. Curran, Valuing Asian and portfolio by conditioning on the geometric mean price, Management Science 40 (1994) 1705-1711] and of [G. Deelstra, J. Liinev, M. Vanmaele, Pricing of arithmetic basket options by conditioning, Insurance: Mathematics & Economics 34 (2004) 55-57] in several ways. We create a framework that allows for a whole class of conditioning random variables which are normally distributed. We moment match not only with a lognormal random variable but also with a log-extended-skew-normal random variable. We also improve the bounds of [G. Deelstra, I. Diallo, M. Vanmaele, Bounds for Asian basket options, Journal of Computational and Applied Mathematics 218 (2008) 215-228]. Numerical results are included and on the basis of our numerical tests, we explain which method we recommend depending on moneyness and time-to-maturity.     

Global attractors for small samples and germs of 3D Navier–Stokes equations

We prove the existence of a global attractor for the generalized semiflow (in the sense of J.M. Ball) on the space of small samples of solutions to the 3D incompressible Navier–Stokes equations. This way to overcome the possible nonuniqueness of solutions is less radical than that of G. Sell and does not provide unique solutions. On the other hand, the existence of the global attractor does not need the unproven hypothesis of continuity of solutions required by Ball. The extension of this approach to the space of germs of solutions is also discussed.     

Equiframed Curves – A Generalization of Radon Curves

Equiframed curves are centrally symmetric convex closed planar curves that are touched at each of their points by some circumscribed parallelogram of smallest area. These curves and their higher-dimensional analogues were introduced by Peczynski and Szarek (1991, Math Proc Cambridge Philos Soc 109: 125–148). Radon curves form a proper subclass of this class of curves. Our main result is a construction of an arbitrary equiframed curve by appropriately modifying a Radon curve. We give characterizations of each type of curve to highlight the subtle difference between equiframed and Radon curves and show that, in some sense, equiframed curves behave dually to Radon curves.     

A new proof of the Gasca–Maeztu conjecture for

In the Chung–Yao construction of poised nodes for bivariate polynomial interpolation [K.C. Chung, T.H. Yao, On lattices admitting unique Lagrange interpolations, SIAM J. Numer. Anal. 14 (1977) 735–743], the interpolation nodes are intersection points of some lines. The Berzolari–Radon construction [L. Berzolari, Sulla determinazione di una curva o di una superficie algebrica e su alcune questioni di postulazione, Lomb. Ist. Rend. 47 (2) (1914) 556–564; J. Radon, Zur mechanischen Kubatur, Monatsh. Math. 52 (1948) 286–300] seems to be more general, since in this case the nodes of interpolation lie (almost) arbitrarily on some lines. In 1982 Gasca and Maeztu conjectured that every poised set allowing the Chung–Yao construction is of Berzolari–Radon type. So far, this conjecture has been confirmed only for polynomial spaces of small total degree n≤4, the result being evident for n≤2 and not hard to see for n=3. For the case n=4 two proofs are known: one of J.R. Busch [J.R. Busch, A note on Lagrange interpolation in , Rev. Un. Mat. Argentina 36 (1990) 33–38], and another of J.M. Carnicer and M. Gasca [J.M. Carnicer, M. Gasca, A conjecture on multivariate polynomial interpolation, Rev. R. Acad. Cienc. Exactas Fís. Nat. (Esp.) Ser. A Mat. 95 (2001) 145–153]. Here we present a third proof which seems to be more geometric in nature and perhaps easier. We also present some results for the case of n=5 and for general n which might be useful for later consideration of the problem.     

Maximal and Cohen–Macaulay projective monomial curves

In this paper we continue the investigation of Cohen–Macaulay projective monomial curves begun in [Les Reid, Leslie G. Roberts, Non-Cohen–Macaulay projective monomial curves, J. Algebra 291 (2005) 171–186]. In the process we introduce maximal curves. Cohen–Macaulay curves are maximal, but not conversely. We show that the number of all curves of degree d that are Cohen–Macaulay grows exponentially, but not as fast as the total number of curves, and also that maximal curves of degree d with sufficiently large embedding dimension relative to d are Cohen–Macaulay.     

Characterization of integrals with respect to arbitrary radon measures by the boundedness indices

The problem of characterization of integrals as linear functionals is considered in the paper. It starts from the familiar results of F. Riesz (1909) and J. Radon (1913) on integral representation of bounded linear functionals by Riemann?CStieltjes integrals on a segment and by Lebesgue integrals on a compact in $ {\mathbb{R}^n} $ , respectively. After works of J. Radon, M. Fréchet, and F. Hausdorff the problem of characterization of integrals as linear functionals took the particular form of the problem of extension of Radon??s theorem from $ {\mathbb{R}^n} $ to more general topological spaces with Radon measures. This problem has turned out difficult and its solution has a long and rich history. Therefore, it may be naturally called the Riesz?CRadon?CFréchet problem of characterization of integrals. The important stages of its solution are connected with such mathematicians as S. Banach, S. Saks, S. Kakutani, P. Halmos, E. Hewitt, R. E. Edwards, N. Bourbaki, V. K. Zakharov, A. V. Mikhalev, et al. In this paper, the Riesz?CRadon?CFr??echet problem is solved for the general case of arbitrary Radon measures on Hausdorff spaces. The solution is given in the form of a general parametric theorem in terms of a new notion of the boundedness index of a functional. The theorem implies as particular cases well-known results of the indicated authors characterizing Radon integrals for various classes of Radon measures and topological spaces.     

Equiframed Curves – A Generalization of Radon Curves

Equiframed curves are centrally symmetric convex closed planar curves that are touched at each of their points by some circumscribed parallelogram of smallest area. These curves and their higher-dimensional analogues were introduced by Peczynski and Szarek (1991, Math Proc Cambridge Philos Soc 109: 125–148). Radon curves form a proper subclass of this class of curves. Our main result is a construction of an arbitrary equiframed curve by appropriately modifying a Radon curve. We give characterizations of each type of curve to highlight the subtle difference between equiframed and Radon curves and show that, in some sense, equiframed curves behave dually to Radon curves.Research supported by a grant from a cooperation between the Deutsche Forschungsgemeinschaft in Germany and the National Research Foundation in South Africa. Parts of this paper were written during a visit to the Department of Mathematics, Applied Mathematics and Astronomy of the University of South Africa.     

Telescopic actions

A group action H on X is called ??telescopic?? if for any finitely presented group G, there exists a subgroup H?? in H such that G is isomorphic to the fundamental group of X/H??. We construct examples of telescopic actions on some CAT[?C1] spaces, in particular on 3 and 4-dimensional hyperbolic spaces. As applications we give new proofs of the following statements: (1) Aitchison??s theorem: Every finitely presented group G can appear as the fundamental group of M/J, where M is a compact 3-manifold and J is an involution which has only isolated fixed points; (2) Taubes?? theorem: Every finitely presented group G can appear as the fundamental group of a compact complex 3-manifold.     

The M/G/1/K blocking formula and its generalizations to state-dependent vacation systems and priority systems

The formula for the blocking probability for the finite capacity M/G/1/K in terms of the steady state occupancy probability distribution of M/G/1 and the system utilization is known [Keilson, J. Royal Statistical Soc. Serie B, 28 (1966) 190–201]. The validity of this relationship is demonstrated for a broad class of state dependent M/G/1 vacation systems and priority systems. New methods are employed which may also be of interest in their own right.This research was conducted while J. Keilson was a Senior Staff Scientist at GTE Laboratories Incorporated.     

A characterization of inverse Radon transform on the Laguerre hypergroup

Let K=[0,∞)×R be the Laguerre hypergroup which is the fundamental manifold of the radial function space for the Heisenberg group. In this note we give another characterization for a subspace of S(K) (Schwartz space) such that the Radon transform Rα on K is a bijection. We show that this characterization is equivalent to that in [M.M. Nessibi, K. Trimèche, Inversion of the Radon transform on the Laguerre hypergroup by using generalized wavelets, J. Math. Anal. Appl. 208 (1997) 337-363]. In addition, we establish an inversion formula of the Radon transform Rα in the weak sense.     

Characterization of radon integrals as linear functionals

The problem of characterization of integrals as linear functionals is considered in this paper. It has its origin in the well-known result of F. Riesz (1909) on integral representation of bounded linear functionals by Riemann?CStieltjes integrals on a segment and is directly connected with the famous theorem of J. Radon (1913) on integral representation of bounded linear functionals by Lebesgue integrals on a compact in ? ⁿ . After the works of J. Radon, M. Fréchet, and F. Hausdorff, the problem of characterization of integrals as linear functionals has been concretized as the problem of extension of Radon??s theorem from ? ⁿ to more general topological spaces with Radon measures. This problem turned out to be difficult, and its solution has a long and abundant history. Therefore, it may be naturally called the Riesz?CRadon?CFréchet problem of characterization of integrals. The important stages of its solution are connected with such eminent mathematicians as S. Banach (1937?C38), S. Saks (1937?C38), S. Kakutani (1941), P. Halmos (1950), E. Hewitt (1952), R. E. Edwards (1953), Yu. V. Prokhorov (1956), N. Bourbaki (1969), H. K¨onig (1995), V. K. Zakharov and A. V. Mikhalev (1997), et al. Essential ideas and technical tools were worked out by A. D. Alexandrov (1940?C43), M. N. Stone (1948?C49), D. H. Fremlin (1974), et al. The article is devoted to the modern stage of solving this problem connected with the works of the authors (1997?C2009). The solution of the problem is presented in the form of the parametric theorems on characterization of integrals. These theorems immediately imply characterization theorems of the above-mentioned authors.     

On the Szegő–Radon projection of monogenic functions

We define a version of the Radon transform for monogenic functions which is based on Szegő kernels. The corresponding Szegő–Radon projection is abstractly defined as the orthogonal projection of a Hilbert module of left monogenic functions onto a suitable closed submodule of functions depending only on two variables. We also establish the inversion formula based on the dual transform.     

Morera theorems via microlocal analysis

We prove Morera theorems for curves in the plane using microlocal analysis. The key is that microlocal smoothness of functions is reflected by smoothness of their Morera integrals on curvestheir Radon transforms. Parallel support theorems for the associated Radon transforms follow from our arguments by a simple correspondence.     

Ratio Limit Theorems for Random Walks and Lévy Processes

In this paper, we study quasi-symmetric random walks and Lévy processes, a property first introduced by C.J. Stone, discuss the -invariant Radon measures for random walks and Lévy processes, and formulate some nice ratio limit theorems which are closely related to -invariant Radon measures. Mathematics Subject Classifications (2000) 60G51, 60G50.Research supported in part by NSFC 10271109.     

A Wiener-Tauberian and a Pompeiu type theorems on the Laguerre hypergroup

A Wiener-Tauberian theorem is proven on the Laguerre hypergroup [M.M. Nessibi, K. Trimèche, Inversion of the Radon transform on the Laguerre hypergroup by using generalized wavelets, J. Math. Anal. Appl. 208 (1997) 337-363]. As consequence of this theorem we establish a Pompeiu type-theorem and we study some of its applications.     

有界域上局部与全局的Radon测度的积分不等式

选择Laplace-Beltrami算子Δ和Green算子G的复合算子Δ◇G为研究对象,首先证明了有界域的局部圆域上作用于齐次A-调和方程解的复合算子Δ◇G的带Radon测度的积分不等式,然后在此基础上得到有界域上全局的Radon积分不等式.     

Explicit Solutions for the Wave Equation on Homogeneous Trees

In this paper we consider the discretized version of the wave equation, in which a manifold is replaced by a homogeneous tree and the time line is replaced by the natural numbers. We give two methods for finding a closed form of the solution. One of these methods is found by first solving the Radon transform of the solution, which has a much simpler form. We also find a simple formula for the Radon transformation of the solution to the heat equation on homogeneous trees.     

A new class of transport distances between measures

We introduce a new class of distances between nonnegative Radon measures in . They are modeled on the dynamical characterization of the Kantorovich-Rubinstein-Wasserstein distances proposed by Benamou and Brenier (Numer Math 84:375–393, 2000) and provide a wide family interpolating between the Wasserstein and the homogeneous -Sobolev distances. From the point of view of optimal transport theory, these distances minimize a dynamical cost to move a given initial distribution of mass to a final configuration. An important difference with the classical setting in mass transport theory is that the cost not only depends on the velocity of the moving particles but also on the densities of the intermediate configurations with respect to a given reference measure γ. We study the topological and geometric properties of these new distances, comparing them with the notion of weak convergence of measures and the well established Kantorovich-Rubinstein-Wasserstein theory. An example of possible applications to the geometric theory of gradient flows is also given. J. Dolbeault and B. Nazaret have been partially supported by the ANR project IFO. The second author has also been partially supported by the ANR project OTARIE. G. Savaré has been partially supported by grants of M.I.U.R., PRIN ’06. Part of this research was carried out while the third author was visiting professor at Ceremade, Université Paris-Dauphine, whose hospitality and support are also gratefully acknowledged.