L—fuzzy Ring and L—fuzzy Module

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Lê cycles and Milnor classes

The purpose of this work is to establish a link between the theory of Chern classes for singular varieties and the geometry of the varieties in question. Namely, we show that if Z is a hypersurface in a compact complex manifold, defined by the complex analytic space of zeroes of a reduced non-zero holomorphic section of a very ample line bundle, then its Milnor classes, regarded as elements in the Chow group of Z, determine the global Lê cycles of Z; and vice versa: The Lê cycles determine the Milnor classes. Morally this implies, among other things, that the Milnor classes determine the topology of the local Milnor fibres at each point of Z, and the geometry of the local Milnor fibres determines the corresponding Milnor classes.     

Lévy processes and quasi-shuffle algebras

We investigate the algebra of repeated integrals of semimartingales. We prove that a minimal family of semimartingales generates a quasi-shuffle algebra. In essence, to fulfil the minimality criterion, first, the family must be a minimal generator of the algebra of repeated integrals generated by its elements and by quadratic covariation processes recursively constructed from the elements of the family. Second, recursively constructed quadratic covariation processes may lie in the linear span of previously constructed quadratic covariation processes and of the family, but may not lie in the linear span of repeated integrals of these. We prove that a finite family of independent Lévy processes that have finite moments generates a minimal family. Key to the proof are the Teugels martingales and a strong orthogonalization of them. We conclude that a finite family of independent Lévy processes forms a quasi-shuffle algebra. We discuss important potential applications to constructing efficient numerical methods for the strong approximation of stochastic differential equations driven by Lévy processes.     

Lévy-type Processes and Besov Spaces

In [6, Théorème VI. 1], it is shown that almost all sample paths of a given stable process (Z_t) of index belong to the Besov spaces with 1 p < . Our aim is to establish a similar result for general Lévy processes (X_t)_{t 0}. It will turn out that if we restrict the paths to compact time intervals (and put them zero elsewhere) then they belong to Besov spaces for a certain choice of parameters s and p. Finally we extend the results obtained for Lévy processes to Markov processes, which are – in a certain sense – comparable with the given Lévy process.     

Sine and cosine series in L φ classes

We study sine and cosine series with monotone with respect to subsequences coefficients. We establish conditions under which their sums belong to classes L _φ .     

Probabilistic and fractal aspects of Lévy trees

We investigate the random continuous trees called Lévy trees, which are obtained as scaling limits of discrete Galton-Watson trees. We give a mathematically precise definition of these random trees as random variables taking values in the set of equivalence classes of compact rooted -trees, which is equipped with the Gromov-Hausdorff distance. To construct Lévy trees, we make use of the coding by the height process which was studied in detail in previous work. We then investigate various probabilistic properties of Lévy trees. In particular we establish a branching property analogous to the well-known property for Galton-Watson trees: Conditionally given the tree below level a, the subtrees originating from that level are distributed as the atoms of a Poisson point measure whose intensity involves a local time measure supported on the vertices at distance a from the root. We study regularity properties of local times in the space variable, and prove that the support of local time is the full level set, except for certain exceptional values of a corresponding to local extinctions. We also compute several fractal dimensions of Lévy trees, including Hausdorff and packing dimensions, in terms of lower and upper indices for the branching mechanism function which characterizes the distribution of the tree. We finally discuss some applications to super-Brownian motion with a general branching mechanism.     

L ∞ Variational Problems with Running Costs and Constraints

Various approaches are used to derive the Aronsson–Euler equations for L ^∞ calculus of variations problems with constraints. The problems considered involve holonomic, nonholonomic, isoperimetric, and isosupremic constraints on the minimizer. In addition, we derive the Aronsson–Euler equation for the basic L ^∞ problem with a running cost and then consider properties of an absolute minimizer. Many open problems are introduced for further study.     

Linear L–Positive Sets and Their Polar Subspaces

In this paper, we define a Banach SNL space to be a Banach space with a certain kind of linear map from it into its dual, and we develop the theory of linear L?Cpositive subsets of Banach SNL spaces with Banach SNL dual spaces. We use this theory to give simplified proofs of some recent results of Bauschke, Borwein, Wang and Yao, and also of the classical Brezis?CBrowder theorem.     

Two Generating Mappings ωL and lL in Complete Lattices TVS and LFTVS

61.IntroductionRecently,weseparatelygavetheconceptofL-fuzzytoPOfoicalvectorspaceanddiscussthecontinuityofL-fuzzylinearorder-homomorphismin[lJ,[2],andweobtainedalotofin-terestingpropertiesinL-fuzzytopologicalvectorsPaces.Butthereisabasicproblem,i.e.whetherthedefinitionofL-fuzzytoPologicalvectorsPacesin[1]isagoodextensionornot,isn,tsolved.Inviewofthis,weintendheretodiscusstwogeneratingmappingscoLandt.L4j'andweprovedthedefinitionofL-fuzzytoPOlogicalvectorspacesisagoodextensioninthe'senseofR…     

Lévy-driven Volterra Equations in Space and Time

We investigate nonlinear stochastic Volterra equations in space and time that are driven by Lévy bases. Under a Lipschitz condition on the nonlinear term, we give existence and uniqueness criteria in weighted function spaces that depend on integrability properties of the kernel and the characteristics of the Lévy basis. Particular attention is devoted to equations with stationary solutions, or more generally, to equations with infinite memory, that is, where the time domain of integration starts at minus infinity. Here, in contrast to the case where time is positive, the usual integrability conditions on the kernel are no longer sufficient for the existence and uniqueness of solutions, but we have to impose additional size conditions on the kernel and the Lévy characteristics. Furthermore, once the existence of a solution is guaranteed, we analyze its asymptotic stability, that is, whether its moments remain bounded when time goes to infinity. Stability is proved whenever kernel and characteristics are small enough, or the nonlinearity of the equation exhibits a fractional growth of order strictly smaller than one. The results are applied to the stochastic heat equation for illustration.     

Realization of Configurations and the Löwner Ellipsoid

It is proved that any subset of an (m-1)-dimensional sphere of volume larger than l(m+ 1) of the volume of the entire sphere contains l+ 1 points forming a regular l-dimensional simplex. As a corollary, it is obtained that, if the exterior of a given m-dimensional filled ellipsoid contains no more than the 1/(m+ 1) fraction of some sphere, then the volume of the ellipsoid is no less than the volume of the corresponding ball. The existence of a pair of points a given spherical distance apart in a set of positive measure is examined.     

A sharp Markov brothers-type inequality in the spaces L ∞ and L 1 on the segment

The inequality between the uniform norm of the derivative of order ? of an algebraic polynomial of degree n and the L ₁-norm of the polynomial itself on a segment are studied. For all ? ≥ (n ? 1)/3, the exact constant and the extremal polynomial are written out.     

Multivariate Bernoulli and Euler polynomials via Lévy processes

By a symbolic method, we introduce multivariate Bernoulli and Euler polynomials as powers of polynomials whose coefficients involve multivariate Lévy processes. Many properties of these polynomials are stated straightforwardly thanks to this representation, which could be easily implemented in any symbolic manipulation system. A very simple relation between these two families of multivariate polynomials is provided.     

Zipf’s law and L. Levin probability distributions

Zipf’s law in its basic incarnation is an empirical probability distribution governing the frequency of usage of words in a language. As Terence Tao recently remarked, it still lacks a convincing and satisfactory mathematical explanation. In this paper I suggest that, at least in certain situations, Zipf’s law can be explained as a special case of the a priori distribution introduced and studied by L. Levin. The Zipf ranking corresponding to diminishing probability appears then as the ordering by growing Kolmogorov complexity. One argument justifying this assertion is the appeal to a recent interpretation by Yu. Manin and M. Marcolli of asymptotic bounds for error-correcting codes in terms of phase transition. In the respective partition function, the Kolmogorov complexity of a code plays the role of its energy.     

Maximal Inequalities for Fractional Lévy and Related Processes

In this article we study processes that are constructed by a convolution of a deterministic kernel with a martingale. A special emphasis is put on the case where the driving martingale is a centred Lévy process, which covers the popular class of fractional Lévy processes. As a main result we show that, under appropriate assumptions on the kernel and the martingale, the maximum process of the corresponding “convoluted martingale” is p-integrable and we derive maximal inequalities in terms of the kernel and of the moments of the driving martingale.