Long-term and blow-up behaviors of exponential moments in multi-dimensional affine diffusions

This paper considers multi-dimensional affine processes with continuous sample paths. By analyzing the Riccati system, which is associated with affine processes via the transform formula, we fully characterize the regions of exponents in which exponential moments of a given process do not explode at any time or explode at a given time. In these two cases, we also compute the long-term growth rate and the explosion rate for exponential moments. These results provide a handle to study implied volatility asymptotics in models where log-returns of stock prices are described by affine processes whose exponential moments do not have an explicit formula.     

The Rogers–Selberg recursions, the Gordon–Andrews identities and intertwining operators

Using the theory of intertwining operators for vertex operator algebras we show that the graded dimensions of the principal subspaces associated to the standard modules for satisfy certain classical recursion formulas of Rogers and Selberg. These recursions were exploited by Andrews in connection with Gordon’s generalization of the Rogers–Ramanujan identities and with Andrews’ related identities. The present work generalizes the authors’ previous work on intertwining operators and the Rogers–Ramanujan recursion. 2000 Mathematics Subject Classification Primary—17B69, 39A13 S. Capparelli gratefully acknowledges partial support from MIUR (Ministero dell’Istruzione, dell’Università e della Ricerca). J. Lepowsky and A. Milas gratefully acknowledge partial support from NSF grant DMS-0070800.     

Real quadrics in C n , complex manifolds and convex polytopes

In this paper, we investigate the topology of a class of non-Kähler compact complex manifolds generalizing that of Hopf and Calabi-Eckmann manifolds. These manifolds are diffeomorphic to special systems of real quadrics C ⁿ which are invariant with respect to the natural action of the real torus (S ¹) ⁿ onto C ⁿ . The quotient space is a simple convex polytope. The problem reduces thus to the study of the topology of certain real algebraic sets and can be handled using combinatorial results on convex polytopes. We prove that the homology groups of these compact complex manifolds can have arbitrary amount of torsion so that their topology is extremely rich. We also resolve an associated wall-crossing problem by introducing holomorphic equivariant elementary surgeries related to some transformations of the simple convex polytope. Finally, as a nice consequence, we obtain that affine non-Kähler compact complex manifolds can have arbitrary amount of torsion in their homology groups, contrasting with the Kähler situation.     

Spherical harmonic analysis on affine buildings

Let be a locally finite regular affine building with root system R. There is a commutative algebra spanned by averaging operators A _λ , λ ∈ P ⁺, acting on the space of all functions f:V _P → , where V _P is in most cases the set of all special vertices of , and P ⁺ is a set of dominant coweights of R. This algebra is studied in [6] and [7] for à _n buildings, and the general case is treated in [15]. In this paper we show that all algebra homomorphisms h: may be expressed in terms of the Macdonald spherical functions. We also provide a second formula for these homomorphisms in terms of an integral over the boundary of . We may regard as a subalgebra of the C ^*-algebra of bounded linear operators on ?²(V _P ), and we write for the closure of in this algebra. We study the Gelfand map , where M ₂= , and we compute M ₂ and the Plancherel measure of . We also compute the ?²-operator norms of the operators A _λ , λ ∈ P ⁺, in terms of the Macdonald spherical functions.     

Comparison theorems for separable wavelet frames

It is known that wavelet frames do not exhibit a Nyquist density. Even so, this paper shows that the affine densities of the sets U×V and S×T affect the frame properties of and . In particular, it is shown that there is a relationship between the densities of the dilation sets U and S and weighted admissibility constants of f and g. This relationship implies a comparison theorem, whereby the affine densities of U×V and S×T are proportional, with proportionality constant depending on the frame bounds and the admissibility constants of f and g. These results are also extended to wavelet frame sequences.     

A two-to-one map and abelian affine difference sets

Let D be an affine difference set of order n in an abelian group G relative to a subgroup N. Set = H \ {1, ω}, where H = G/N and . Using D we define a two-to-one map g from to N. The map g satisfies g(σ ^m ) = g(σ) ^m and g(σ) = g(σ ⁻¹) for any multiplier m of D and any element σ ∈ . As applications, we present some results which give a restriction on the possible order n and the group theoretic structure of G/N.      

Semilocal convergence for Halley’s method under weak Lipschitz condition

The semilocal convergence properties of Halley’s method for nonlinear operator equations are studied under the hypothesis that the second derivative satisfies some weak Lipschitz condition. The method employed in the present paper is based on a family of recurrence relations which will be satisfied by the involved operator. An application to a nonlinear Hammerstein integral equation of the second kind is provided.     

3-齐次群与旗传递5-（v，k，3）设计

旗传递t-设计的分类是代数组合学的一个重要课题．本文主要讨论了旗传递5-（v,k,3）设计．由P．J．Cameron和C．E．Praeger的结论可知,此时设计的自同构群是3-齐次群．本文利用3-齐次群的分类,证明了设计的自同构群不能是仿射型群．