Transformation of hypersingular integrals and black-box cubature

In this paper, we will consider hypersingular integrals as they arise by transforming elliptic boundary value problems into boundary integral equations. First, local representations of these integrals will be derived. These representations contain so-called finite-part integrals. In the second step, these integrals are reformulated as improper integrals. We will show that these integrals can be treated by cubature methods for weakly singular integrals as they exist in the literature.

     

Real Principal Value Integrals

 We study principal value integrals of multi-valued differential forms on compact spaces, as introduced by Langlands. Using resolutions of singularities we extend Langlands definition to the case in which the differential form may have no normal crossings. We show by an example that for non-compact spaces the principal value integral associated to a compactification may depend on the compactification. Principal value integrals appear as residues of poles of distributions and as coefficients of asymptotic expansions of oscillating integrals and fibre integrals.     

Symmetry Classification for Jackson Integrals Associated with the Root System BC n

The Jackson integrals associated with the non-reduced root system are defined as multiple sums which are generalization of the Bailey's very-well-poised ₆₆ sum. They are classified by the number of their parameters when they can be expressed as a product of the Jacobi elliptic theta functions. The sums which appear in the classification list coincide with those investigated individually by Gustafson and van Diejen.     

A functional non-central limit theorem for multiple-stable processes with long-range dependence

A functional limit theorem is established for the partial-sum process of a class of stationary sequences which exhibit both heavy tails and long-range dependence. The stationary sequence is constructed using multiple stochastic integrals with heavy-tailed marginal distribution. Furthermore, the multiple stochastic integrals are built upon a large family of dynamical systems that are ergodic and conservative, leading to the long-range dependence phenomenon of the model. The limits constitute a new class of self-similar processes with stationary increments. They are represented by multiple stable integrals, where the integrands involve the local times of intersections of independent stationary stable regenerative sets.     

Real Principal Value Integrals

 We study principal value integrals of multi-valued differential forms on compact spaces, as introduced by Langlands. Using resolutions of singularities we extend Langlands definition to the case in which the differential form may have no normal crossings. We show by an example that for non-compact spaces the principal value integral associated to a compactification may depend on the compactification. Principal value integrals appear as residues of poles of distributions and as coefficients of asymptotic expansions of oscillating integrals and fibre integrals. (Received 11 November 1999)     

Sums over Graphs and Integration over Discrete Groupoids

We show that sums over graphs such as appear in the theory of Feynman diagrams can be seen as integrals over discrete groupoids. From this point of view, basic combinatorial formulas of the theory of Feynman diagrams can be interpreted as pull-back or push-forward formulas for integrals over suitable groupoids.     

Complex valued multiparameter stochastic integrals

In this paper we will consider the properties of various stochastic integrals over a complex valued, two parameter Wiener process. Integrals of this type exhibit pleasant features which do not appear within the real framework. They are stable under approximation and viewed in relation with analytic functions, they typically satisfy an ordinary chain rule. This in turn gives rise to several nice representation formulas.     

On the symmetric fourth powerL-function of GL2

In this paper I introduce a global Rankin-Selberg integral representing the symmetric fourth powerL-function for GL(2). I show that the global integral is factorizable and compute the local unramified integrals. Finally, I also study some other properties of the local nonarchimedean integrals.     

Triple product L-functions and quantum chaos on SL(2, ℂ)

Let Y be an arithmetic hyperbolic 3-manifold. We establish a link between quantum unique ergodicity for sections of automorphic vector bundles on Y and subconvexity for the triple product L-function, which extends a result of Watson in the case of hyperbolic 2-manifolds. The proof combines the representation theoretic microlocal lift for bundles developed by Bunke and Olbrich with the triple product formula of Ichino. A key step is determining the asymptotic behaviour of the local integrals at complex places that appear in Ichino’s formula.     

The interpretation of fuzzy integrals and their application to fuzzy systems

Fuzzy integrals, in general, and Sugeno integrals, in particular, are well known aggregation operators. They can be used in a great variety of decision making applications. Nevertheless, their use is not easy as their interpretation is not straightforward. In this paper we study the interpretation of fuzzy integrals, focusing on Sugeno ones, and we show their application to fuzzy inference systems when the rules are not independent.     

The Hölder exponent of some Fourier series

In this paper we study the local regularity of fractional integrals of Fourier series using several definitions of the Hölder exponent. We especially consider series coming from fractional integrals of modular forms. Our results show that in general, cusp forms give rise to pure fractals (as opposed to multifractals). We include explicit examples and computer plots.     

一类哈密顿系统的Abel积分之比单调性的判据

研究了一类哈密顿系统的两个Abel积分比值的单调性的条件,指出这个单调性条件可由文中给出的两个判定函数直接确定.     

Symmetry Classification for Jackson Integrals Associated with Irreducible Reduced Root Systems

We state certain product formulae for Jackson integrals associated with irreducible reduced root systems. The Jackson integral is defined here as a sum over any full-rank sublattice of the coweight lattice for the root system. In particular, a Weyl group symmetry classification of the Jackson integrals is done when they have an expression of a product of the Jacobi elliptic theta functions. Most of the product formulae investigated by Aomoto, Macdonald and Gustafson appear in the list of classifications. A new product formula for an F ₄ root system is included in it.     

On some integrals of Glauert’s type

Some principal value integrals of Glauert’s type are calculated analytically. These integrals appear when dealing with the hydrodynamic impact problem of a two-dimensional wedge onto a liquid free surface in the potential theory.     

Limits of elliptic hypergeometric integrals

In Ann. Math., to appear, 2008, the author proved a number of multivariate elliptic hypergeometric integrals. The purpose of the present note is to explore more carefully the various limiting cases (hyperbolic, trigonometric, rational, and classical) that exist. In particular, we show (using some new estimates of generalized gamma functions) that the hyperbolic integrals (previously treated as purely formal limits) are indeed limiting cases. We also obtain a number of new trigonometric (q-hypergeometric) integral identities as limits from the elliptic level. The author was supported in part by NSF Grant No. DMS-0401387.     

Local first integrals of differential systems and diffeomorphisms

In this paper using theory of linear operators and normal forms we generalize a result of Poincaré [11] about the non-existence of local first integrals for systems of differential equations in a neighbourhood of a singular point. As an application of the generalized result, and under more weak conditions we obtain a result of Furta [8] about local first integrals of semi-quasi-homogeneous systems. Moreover, for diffeomorphisms and periodic differential systems we give definitions of their first integrals, and generalize the previous results about systems of differential equations to diffeomorphisms in a neighbourhood of a fixed point and to periodic differential systems in a neighbourhood of a constant solution.     

A unique graph of minimal elastic energy

Nonlinear functionals that appear as a product of two integrals are considered in the context of elastic curves of variable length. A technique is introduced that exploits the fact that one of the integrals has an integrand independent of the derivative of the unknown. Both the linear and the nonlinear cases are illustrated. By lengthening parameterized curves it is possible to reduce the elastic energy to zero. It is shown here that for graphs this is not the case. Specifically, there is a unique graph of minimal elastic energy among all graphs that have turned 90 degrees after traversing one unit.

     

Curvature estimates for the Ricci flow II

In this paper we present several curvature estimates and convergence results for solutions of the Ricci flow, including the volume normalized Ricci flow and the normalized Kähler-Ricci flow. The curvature estimates depend on smallness of certain local space-time integrals of the norm of the Riemann curvature tensor, while the convergence results require finiteness of space-time integrals of this norm. These results also serve as characterization of blow-up singularities.     

一类Hamilton系统的Abel积分比值单调性的判别条件

用直接计算的方法对一类Hamilton系统的两个Abel积分比值的单调性进行讨论,指出该单词性条件可由两个判定函数直接确定．     

New identities of differential operators from orbital integrals on

We derive identities of differential operators on complex general linear groups which appear in the differential equations satisfied by weighted orbital integrals. These identities stem from and have applications to comparisons of metaplectic coverings.