In this article, we prove that the inverse of the Malliavin matrix belongs to L(Ω,P) for a class of degenerate stochastic differential equation(SDE). The conditions required are similar to Hörmander’s bracket condition, but we don’t need all coefficients of the SDE are smooth. Furthermore, we obtain a locally uniform estimate for the Malliavin matrix and a gradient estimate. We also prove that ...

Let I = (x1 , . . . , xq ) be a square-free monomial ideal of a polynomial ring K[x1, . . . , xn] over an arbitrary field K and let A be the incidence matrix with column vectors v1, . . . , vq. We will establish some connections between algebraic properties of certain graded algebras associated to I and combinatorial optimization properties of certain polyhedrons and clutters associated to A an...

We give an account of some results, both old and new, about any n× n Markov matrix that is embeddable in a one-parameter Markov semigroup. These include the fact that its eigenvalues must lie in a certain region in the unit ball. We prove that a well-known procedure for approximating a non-embeddable Markov matrix by an embeddable one is optimal in a certain sense.

The concept of residual intersection, introduced by Artin and Nagata [l] in 1972, is a fruitful generalization of linkage as the following two examples attest. Let I be a strongly Cohen-Macaulay ideal in a CohenMacaulay local ring R. If Z satisfies the condition (G,), then Huneke [6, Proposition 4.31 has proved that the extended Rees algebra R[It, t‘1 is defined by an ideal which is obtained fr...

The determination of trace rare-earth elements (REEs) can be used for the assessment of environmental pollution, and is of great significance to the study of toxicity and toxicology in animals and plants. N, N, N', N'-tetraoctyl diglycolamide (TODGA) is an environmental friendly extractant that is highly selective to REEs. In this study, an analytical method was developed for the simultaneous d...

We show a reduction of Hilbert’s tenth problem to the solvability of the matrix equation A1 1 A i2 2 · · ·A ik k = Z over non-commuting integral matrices, where Z is the zero matrix, thus proving that the solvability of the equation is undecidable. This is in contrast to the case whereby the matrix semigroup is commutative in which the solvability of the same equation was shown to be decidable ...

A two-way subtable sum model is a statistical model for two-way contingency tables such that sufficient statistics for the model are row sums, column sums and an additional constraint that the sum of a subtable is also fixed. From a statistical viewpoint this model is related to a block interaction model, a twoway change-point model proposed by Hirotsu, and the quasi-independence model for inco...

This expository paper contains history, definitions, constructions, and the basic properties of Rees valuations of ideals. A section is devoted to one-fibered ideals, that is, ideals with only one Rees valuation. Cutkosky [5] proved that there exists a two-dimensional complete Noetherian local integrally closed domain in which no zero-dimensional ideal is one-fibered. However, no concrete ring ...