Explore the vast range of titles available.

A quantitative sharp form of the classical isoperimetric inequality is proved, thus giving a positive answer to a conjecture by Hall.

Schoenberg (1938) identified the class of positive definite radial (or isotropic) functions φ:Rd↦R\\varphi :\\mathbb {R}^d\\mapsto \\mathbb {R}, φ(0)=1\\varphi ({\\textbf {0}})=1, as having a representation φ(x)=∫R+Ωd(tu)Gd(du)\\varphi ({\\textbf {x}}) = \\int _{\\mathbb {R}_+}\\Omega _d(tu)\\,G_d(\\mathrm {d} u), t=‖x‖t=\\|{\\textbf {x}}\\|, for some uniquely identified probability measure GdG_d on R+\\mathbb {R}_+ and Ωd(t)=E(eit⟨e1,η⟩)\\Omega _d(t) = {\\mathrm {E}} ({\\mathrm {e}} ^{it\\langle {\\textbf {e}} _1, \\mathbf {\\scriptstyle \\eta } \\rangle }), where η\\mathbf {\\eta } is a vector uniformly distributed on the unit spherical shell Sd−1⊂Rd\\mathbb {S} ^{d-1} \\subset \\mathbb {R}^d and e1{\\textbf {e}}_1 is a fixed unit vector. Call such GdG_d a d-Schoenberg measure, and let Φd\\Phi _d denote the class of all functions f:R+↦Rf: \\mathbb {R}_+ \\mapsto \\mathbb {R} for which such a dd-dimensional radial function φ\\varphi exists with f(t)=φ(x)f(t) = \\varphi ({\\textbf {x}} ) for t=‖x‖t=\\| {\\textbf {x}}\\|. Mathéron (1965) introduced operators I~{\\widetilde {I}} and D~{\\widetilde {D}}, called Montée and Descente, that map suitable f∈Φdf\\in \\Phi _d into Φd′\\Phi _{d’} for some different dimension d′d’: Wendland described such mappings as dimension walks. This paper characterizes Mathéron’s operators in terms of Schoenberg measures and describes functions, even in the class Φ∞\\Phi _\\infty of completely monotone functions, for which neither I~f{\\widetilde {I}} f nor D~f{\\widetilde {D}} f is well defined. Because f∈Φdf\\in \\Phi _d implies f∈Φd′f\\in \\Phi _{d’} for d′>dd’>d, any f∈Φdf\\in \\Phi _d has a d′d’-Schoenberg measure Gd′G_{d’} for 1≤d′>d1\\le d’>d and finite d≥2d\\ge 2. This paper identifies Gd′G_{d’} in terms of GdG_d via another ‘dimension walk’ relating the Fourier transforms Ωd′\\Omega _{d’} and Ωd\\Omega _d that reflect projections on Rd′\\mathbb {R} ^{d’} within Rd\\mathbb {R} ^d. A study of the Euclid hat function shows the indecomposability of Ωd\\Omega _d.

We establish inversion formulas of the so-called filtered back-projection type to recover a function supported in the ball in even dimensions from its spherical means over spheres centered on the boundary of the ball. We also find several formulas to recover initial data of the form (f, 0) (or (0, g)) for the free space wave equation in even dimensions from the trace of the solution on the boundary of the ball, provided that the initial data has support in the ball.

We combine Stein's method with a version of Malliavin calculus on the Poisson space. As a result, we obtain explicit Berry–Esséen bounds in Central limit theorems (CLTs) involving multiple Wiener–Itô integrals with respect to a general Poisson measure. We provide several applications to CLTs related to Ornstein–Uhlenbeck Lévy processes.

We introduce spaces of exponential constructible functions in the motivic setting for which we construct direct image functors in the absolute and relative settings. This allows us to define a motivic Fourier transformation for which we get various inversion statements. We also define spaces of motivic Schwartz-Bruhat functions on which motivic Fourier transformation induces isomorphisms. Our motivic integrals specialize to nonarchimedean integrals. We give a general transfer principle comparing identities between functions defined by exponential integrals over local fields of characteristic zero, resp. of positive characteristic, having the same residue field. We also prove new results about p-adic integrals of exponential functions and stability of this class of functions under p-adic integration.

Cianchi et al prove a sharp quantitative version of the isoperimetric inequality in the space R^sup n^ endowed with the Gaussian measure. The isoperimetric inequality in Gauss space asserts that among all subsets of R^sup n^ with prescribed Gaussian measure, half-spaces have the least Gaussian perimeter. They also provide a geometric proof of the Gaussian isoperimetric theorem.

The question of existence and properties of stationary solutions to Langevin equations driven by noise processes with stationary increments is discussed, with particular focus on noise processes of pseudo-moving-average type. On account of the Wold-Karhunen decomposition theorem, such solutions are, in principle, representable as a moving average (plus a drift-like term) but the kernel in the moving average is generally not available in explicit form. A class of cases is determined where an explicit expression of the kernel can be given, and this is used to obtain information on the asymptotic behavior of the associated autocorrelation functions, both for small and large lags. Applications to Gaussian-and Lévy-driven fractional Ornstein-Uhlenbeck processes are presented. A Fubini theorem for Lévy bases is established as an element in the derivations.

We deal with backward stochastic differential equations with time delayed generators. In this new type of equation, a generator at time t can depend on the values of a solution in the past, weighted with a time delay function, for instance, of the moving average type. We prove existence and uniqueness of a solution for a sufficiently small time horizon or for a sufficiently small Lipschitz constant of a generator. We give examples of BSDE with time delayed generators that have multiple solutions or that have no solutions. We show for some special class of generators that existence and uniqueness may still hold for an arbitrary time horizon and for arbitrary Lipschitz constant. This class includes linear time delayed generators which we study in more detail. We are concerned with different properties of a solution of a BSDE with time delayed generator, including the inheritance of boundedness from the terminal condition, the comparison principle, the existence of a measure solution and the BMO martingale property. We give examples in which they may fail.

In this paper, we use the framework of functions of bounded variation and the coarea formula to give an explicit computation for the expectation of the perimeter of excursion sets of shot noise random fields in dimension n ≥ 1. This will then allow us to derive the asymptotic behavior of these mean perimeters as the intensity of the underlying homogeneous Poisson point process goes to infinity. In particular, we show that two cases occur: we have a Gaussian asymptotic behavior when the kernel function of the shot noise has no jump part, whereas the asymptotic is non-Gaussian when there are jumps.

Fractional derivatives can be used to model time delays in a diffusion process. When the order of the fractional derivative is distributed over the unit interval, it is useful for modeling a mixture of delay sources. In some special cases a distributed order derivative can be used to model ultra-slow diffusion. We extend the results of Baeumer and Meerschaert in the single order fractional derivative case to the distributed order fractional derivative case. In particular, we develop strong analytic solutions of distributed order fractional Cauchy problems.