Intersection multiplicities over Gorenstein rings

Let R be a complete local ring of dimension d over a perfect field of prime characteristic p, and let M be an R-module of finite length and finite projective dimension. S. Dutta showed that the equality holds when the ring R is a complete intersection or a Gorenstein ring of dimension at most 3. We construct a module over a Gorenstein ring R of dimension five for which this equality fails to hold. This then provides an example of a nonzero Todd class , and of a bounded free complex whose local Chern character does not vanish on this class. Received: February 11, 1999     

Mixed multiplicities for arbitrary ideals and generalized Buchsbaum-Rim multiplicities

We introduce first the notion of mixed multiplicities for arbitraryideals in a local d-dimensional Noetherian ring (A,m) which,in some sense, generalizes the concept of mixed multiplicitiesfor m-primary ideals. We also generalize Teissier's productformula for a set of arbitrary ideals and extend the notionof the Buchsbaum-Rim multiplicity (BR-multiplicity) of a submoduleof a free module to the case where the submodule no longer hasfinite colength. For a submodule M of A^p, we introduce a sequenceof multiplicities e^k_BR (M), k = 0, . . . , d + p – 1 whichin the case of an ideal (p = 1) coincides with the multiplicitysequence c0(I,A), . . . , c_d(I,A) defined for an arbitrary idealI of A by Achilles and Manaresi. In the case where M has finitecolength in Ap and is totally decomposable, we prove that ourBR-multiplicity sequence essentially falls into the standardBR-multiplicity of M.     

An additive formula for Samuel multiplicities on Hilbert spaces of analytic functions

We establish a short exact sequence to relate the germ model of invariant subspaces of a Hilbert space of vector-valued analytic functions and the sheaf model of the corresponding coinvariant subspaces. As a consequence we obtain an additive formula for Samuel multiplicities. As an application, we give a different proof for a formula relating the fibre dimension and the Samuel multiplicity which is first proved in Fang (2005) [11]. The feature of the new proof is that the analytic arguments in Fang (2005) [11] are now subsumed by algebraic machinery.     

Intersection bodies and generalized cosine transforms

The paper is focused on intimate connection between geometric properties of intersection bodies in convex geometry and generalized cosine transforms in harmonic analysis. A new concept of λ-intersection body, that unifies some known classes of geometric objects, is introduced. A parallel between trace theorems in function theory, restriction onto lower-dimensional subspaces of the spherical Radon transforms and the generalized cosine transforms, and sections of λ-intersection bodies is established. New integral formulas for different classes of cosine transforms are obtained and examples of λ-intersection bodies are given. We also revisit some known facts in this area and give them new simple proofs.     

Intersection numbers of families of ideals

We study the intersection number of families of tall ideals. We show that the intersection number of the class of analytic P-ideals is equal to the bounding number ${\mathfrak{b}}$ , the intersection number of the class of all meager ideals is equal to ${\mathfrak{h}}$ and the intersection number of the class of all F _σ ideals is between ${\mathfrak{h}}$ and ${\mathfrak{b}}$ , consistently different from both.     

On multiplicities of simple subquotients in generalized Verma modules

We reduce the problem on multiplicities of simple subquotients in an -stratified generalized Verma module to the analogous problem for classical Verma modules.     

Tjurina numbers of plane curve singularities whose multiplicities are three and four

We describe Tjurina numbers of irreducible plane curve singularities whose multiplicities are three and four. Received: 4 April 2005     

More generalized packing numbers

The packing number P(r,k,v) is defined to be the minimum number of non-empty subsets of cardinality ?k needed to contain each element of a set of cardinality v exactly r times, with no pair of distinct elements occuring more than once. The packing numbers P(3,4,v), P(5,3,v) and P(6,3,v) are completely determined and some other packing numbers determined. Tables of known packing numbers are given.     

New identities involving generalized Fibonacci and generalized Lucas numbers

This paper presents two new identities involving generalized Fibonacci and generalized Lucas numbers. One of these identities generalize the two well-known identities of Sury and Marques which are recently developed. Some other interesting identities involving the famous numbers of Fibonacci, Lucas, Pell and Pell-Lucas numbers are also deduced as special cases of the two derived identities. Performing some mathematical operations on the introduced identities yield some other new identities involving generalized Fibonacci and generalized Lucas numbers.     

A generalization of generalized Fibonacci and generalized Pell numbers

This paper is concerned with developing a new class of generalized numbers. The main advantage of this class is that it generalizes the two classes of generalized Fibonacci numbers and generalized Pell numbers. Some new identities involving these generalized numbers are obtained. In addition, the two well-known identities of Sury and Marques which are recently developed are deduced as special cases. Moreover, some other interesting identities involving the celebrated Fibonacci, Lucas, Pell and Pell–Lucas numbers are also deduced.     

Rigidity and holomorphic Segre transversality for holomorphic Segre maps

Let and denote the complexifications of Heisenberg hypersurfaces in and , respectively. We show that non-degenerate holomorphic Segre mappings from into with possess a partial rigidity property. As an application, we prove that the holomorphic Segre non-transversality for a holomorphic Segre map from into with propagates along Segre varieties. We also give an example showing that this propagation property of holomorphic Segre transversality fails when N > 2n − 2.     

Certain summation formulas involving harmonic numbers and generalized harmonic numbers

Harmonic numbers and generalized harmonic numbers have been studied since the distant past and involved in a wide range of diverse fields such as analysis of algorithms in computer science, various branches of number theory, elementary particle physics and theoretical physics. Here we aim at presenting further interesting identities about certain finite or infinite series involving harmonic numbers and generalized harmonic numbers by applying an algorithmic method to a known summation formula for the hypergeometric function ₅F₄(1).     

Quantum coin flipping,qubit measurement,and generalized Fibonacci numbers

Theoretical and Mathematical Physics - The problem of Hadamard quantum coin measurement in $$n$$ trials, with an arbitrary number of repeated consecutive last states, is formulated in terms of...     

广义Bernoulli数和广义高阶Bernoulli数

定义了广义Bernoulli数和广义高阶Bernoulli数，建立了它们的递推公式和有关性质，从而推广了Bernoulli数和高阶Bernoulli数。     

Divisibility of generalized Catalan numbers

We define a q generalization of weighted Catalan numbers studied by Postnikov and Sagan, and prove a result on the divisibility by p of such numbers when p is a prime and q its power.