The Lê varieties,I

Summary For a complex polynomial,f:( ⁿ⁺¹ ,0) (, 0), with a singular set of complex, dimensions at the origin, we define a sequence of varieties—the Lê varieties, _f ^(k) , off at 0. The multiplicities of these varieties, _f ^(k) , generalize the Milnor number for an isolated singularity. In particular, we show that ifsn-2, the Milnor, fibre off is obtained fromB ²ⁿ by successively attaching _f ^{(n – k)} k-handles, wheren-skn Ifs=n-1, the Milnor fibre off is obtained from a2n-manifold with the homotopy type of a bouquet of _f ^{(n – 1)} circles by successively attaching _f ^{(n – k)} k-handles, where 2kn.The author is a National Science Foundation, Postdoctoral Research Fellow supported by grant # DMS-8807216     

The structure of L*-inverse semigroups

The concepts of L*-inverse semigroups and left wreath products of semigroups are introduced. It is shown that the L*-inverse semigroup can be described as the left wreath product of a type A semigroupΓand a left regular band B together with a mapping which maps the semigroupΓinto the endomorphism semigroup End(B). This result generalizes the structure theorem of Yamada for the left inverse semigroups in the class of regular semigroups. We shall also provide a constructed example for the L*-inverse semigroups by using the left wreath products.     

The Lévy Swap Market Model

Models driven by Lévy processes are attractive since they allow for better statistical fitting than classical diffusion models. The dynamics of the forward swap rate process is derived in a semimartingale setting and a Lévy swap market model is introduced. In order to guarantee positive rates, the swap rates are modelled as ordinary exponentials. The model starts with the most distant rate, which is driven by a non‐homogeneous Lévy process. Via backward induction the remaining swap rates are constructed such that they become martingales under the corresponding forward swap measures. Finally it is shown how swaptions can be priced using bilateral Laplace transforms.     

The L ∞-Stokes semigroup in exterior domains

The Stokes semigroup on a bounded domain is an analytic semigroup on spaces of bounded functions as was recently shown by the authors based on an a priori L ^∞-estimate for solutions to the linear Stokes equations. In this paper, we extend our approach to exterior domains and prove that the Stokes semigroup is uniquely extendable to an analytic semigroup on spaces of bounded functions.     

The multifractal nature of Lévy processes

We show that the sample paths of most Lévy processes are multifractal functions and we determine their spectrum of singularities. Received: 21 February 1997 / Revised version: 27 July 1998     

The Coding Complexity of Lévy Processes

We introduce a new coding scheme for general real-valued Lévy processes and control its performance with respect to L _p [0,1]-norm distortion under different complexity constraints. We also establish lower bounds that prove the optimality of our coding scheme in many cases.      

The Bradley–Terry condition is L1-testable

We provide an algorithm with constant running time that given a weighted tournament $T$, distinguishes with high probability of success between the cases that $T$ can be represented by a Bradley–Terry model, or cannot even be approximated by one. The same algorithm tests whether the corresponding Markov chain is reversible.     

The fractional dimensional theory in Lüroth expansion

It is well known that every x ∈ (0, 1] can be expanded to an infinite Lüroth series in the form of

The Lévy-Itô decomposition in free probability

In this paper we prove the free analog of the Lévy-Itô decomposition for Lévy processes. A significant part of the proof consists of introducing free Poisson random measures, proving their existence and developing a theory of integration with respect to such measures. The existence of free Poisson random measures also yields, via the free Lévy-Itô decomposition, an alternative proof of the general existence of free Lévy processes (in law).MaPhySto – The Danish National Research Foundation Network in Mathematical Physics and StochasticsSupported by the Danish Natural Science Research CouncilMathematics Subject Classification (2000): Primary 46L54; Secondary 60G20, 60G57Acknowledgement We are grateful to the referee for many helpful remarks.     

The Jacobi Field of a Lévy Process

We derive an explicit formula for the Jacobi field that is acting in an extended Fock space and corresponds to an ( -valued) Lévy process on a Riemannian manifold. The support of the measure of jumps in the Lévy–Khintchine representation for the Lévy process is supposed to have an infinite number of points. We characterize the gamma, Pascal, and Meixner processes as the only Lévy process whose Jacobi field leaves the set of finite continuous elements of the extended Fock space invariant.     

The Hausdorff Dimension of Operator Semistable Lévy Processes

Let X={X(t)} _t≥0 be an operator semistable Lévy process in ? ^d with exponent E, where E is an invertible linear operator on ? ^d and X is semi-selfsimilar with respect to E. By refining arguments given in Meerschaert and Xiao (Stoch. Process. Appl. 115, 55–75, 2005) for the special case of an operator stable (selfsimilar) Lévy process, for an arbitrary Borel set B??₊ we determine the Hausdorff dimension of the partial range X(B) in terms of the real parts of the eigenvalues of E and the Hausdorff dimension of B.     

The Most Visited Sites of Certain Lévy Processes

Let X be a symmetric Lévy process with

The efficiency of approximating real numbers by Lüroth expansion

For any x ∈ (0, 1], let $$x = {1 \over {{d_1}}} + {1 \over {{d_1}({d_1} - 1){d_2}}} + \ldots + {1 \over {{d_1}({d_1} - 1) \ldots {d_{n - 1}}({d_{n - 1}} - 1){d_n}}} + \ldots $$ be its Lüroth expansion. Denote by P _n (x)/Q _n (x) the partial sum of the first n terms in the above series and call it the nth convergent of x in the Lüroth expansion. This paper is concerned with the efficiency of approximating real numbers by their convergents {P _n (x)/Q _n (x)} _n?1 in the Lüroth expansion. It is shown that almost no points can have convergents as the optimal approximation for infinitely many times in the Lüroth expansion. Consequently, Hausdorff dimension is introduced to quantify the set of real numbers which can be well approximated by their convergents in the Lüroth expansion, namely the following Jarník-like set: {x ∈ (0, 1]: |x ? P _n (x)/Q _n (x)| < 1/Q _n (x) ^ν+1 infinitely often} for any ν ? 1.     

The L（3,2, 1）-labeling on Bipartite Graphs

An L（3, 2, 1）-labeling of a graph G is a function from the vertex set V（G） to the set of all nonnegative integers such that ｜f（u）-f（v）｜≥3 if dG（u,v） = 1, ｜f（u）-f（v）｜≥2 if dG（u,v） = 2, and ｜f（u）-f（v）｜≥1 if dG（u,v） = 3. The L（3, 2,1）-labeling problem is to find the smallest number λ3（G） such that there exists an L（3, 2,1）-labeling function with no label greater than it. This paper studies the problem for bipartite graphs. We obtain some bounds of λ3 for bipartite graphs and its subclasses. Moreover, we provide a best possible condition for a tree T such that λ3（T） attains the minimum value.     

The Cauchy Problem for Nonlinear Parabolic Equations with Lévy Laplacian

A solution to the Cauchy problem for a rather general class of nonlinear parabolic equations involving the infinite-dimensional Laplacian Δ_L of the form , where f is a real function defined on R³ is presented. Mathematics Subject Classifications (2000) 35R15, 46G05.     

The polynomial Daugavet property for atomless L 1(μ)-spaces

For any atomless positive measure μ, the space L ₁(μ) has the polynomial Daugavet property, i.e., every weakly compact continuous polynomial ${P:L_1(\mu)\longrightarrow L_1(\mu)}For any atomless positive measure μ, the space L ₁(μ) has the polynomial Daugavet property, i.e., every weakly compact continuous polynomial P:L₁(m)? L₁(m){P:L_1(\mu)\longrightarrow L_1(\mu)} satisfies the Daugavet equation ||Id + P||=1 + ||P||{\|{\rm Id} + P\|=1 + \|P\|}. The same is true for the vector-valued spaces L ₁(μ, E), μ atomless, E arbitrary.     

The ergodicity of stochastic partial differential equations with Lévy jump

In this article, the authors prove the uniqueness in law of a class of stochastic equations in infinite dimension, then we apply it to establish the existence and uniqueness of invariant measure of the generalized stochastic partial differential equation perturbed by Lévy process.     

The H?lder-Poincaré duality for L q,p -cohomology

We prove the following version of Poincaré duality for reduced L _q,p -cohomology: For any 1 < q, p < ∞, the L _q,p -cohomology of a Riemannian manifold is in duality with the interior L _p',q'-cohomology for 1/p + 1/p′ = 1/q + 1/q′ = 1.     

The centralizer of an element in a Lie algebra of type L

In this paper, we investigate the Lie algebra L(A, α, δ) of type L and obtain the respective sufficient conditions for L(A, α,δ) to be semisimple, and for Z(ω) = Fω as well, where 0 ≠ ω ∈ L(A, α, δ) and Z(ω) is the centralizer of ω.     

The Cauchy problem for the wave equation with Lévy Laplacian

We present the solution of the Cauchy problem (the initial-value problem in the whole space) for the wave equation with infinite-dimensional Lévy Laplacian Δ _L , $$ \frac{{\partial ^2 U(t,x)}} {{\partial t^2 }} = \Delta _L U(t,x) $$ in two function classes, the Shilov class and the Gâteaux class.