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     

Une surface de Voss à 6 dimensions dont le sous-espace osculateur est à 9 dimensions

Sans résumé à M. Enrico Bompiani pour un Jubilé scieutificue.     

ω-Gorenstein injective modules and dimensions

Gorenstein injective modules and dimensions have been studied extensively by many authors. In this paper, we investigate Gorenstein injective modules and dimensions relative to a Wakamatsu tilting module.     

On differential–algebraic equations in infinite dimensions

We consider a class of differential–algebraic equations (DAEs) with index zero in an infinite dimensional Hilbert space. We define a space of consistent initial values, which lead to classical continuously differential solutions for the associated DAE. Moreover, we show that for arbitrary initial values we obtain mild solutions for the associated problem. We discuss the asymptotic behaviour of solutions for both problems. In particular, we provide a characterisation for exponential stability and exponential dichotomies in terms of the spectrum of the associated operator pencil.     

Generalized Ginzburg–Landau equations in high dimensions

In this work, we study critical points of the generalized Ginzburg–Landau equations in dimensions \(n\ge 3\) which satisfy a suitable energy bound, but are not necessarily energy-minimizers. When the parameter in the equations tend to zero, such solutions are shown to converge to singular n-harmonic maps into spheres, and the convergence is strong away from a finite set consisting (1) of the infinite energy singularities of the limiting map, and (2) of points where bubbling off of finite energy n-harmonic maps could take place. The latter case is specific to dimensions \({>}2\). We also exhibit a criticality condition satisfied by the limiting n-harmonic maps which constrains the location of the infinite energy singularities. Finally we construct an example of non-minimizing solutions to the generalized Ginzburg–Landau equations satisfying our assumptions.     

The Elekes–Szabó Theorem in four dimensions

Let F ∈ C[x, y, s, t] be an irreducible constant-degree polynomial, and let A,B,C,D ? C be finite sets of size n. We show that F vanishes on at most O(n^8/3) points of the Cartesian product A × B × C × D, unless F has a special group-related form. A similar statement holds for A,B,C,D of unequal sizes, with a suitably modified bound on the number of zeros. This is a four-dimensional extension of our recent improved analysis of the original Elekes–Szabó theorem in three dimensions. We give three applications: an expansion bound for three-variable real polynomials that do not have a special form, a bound on the number of coplanar quadruples on a space curve that is neither planar nor quartic, and a bound on the number of four-point circles on a plane curve that has degree at least five.     

Symmetric finite volume schemes for eigenvalue problems in arbitrary dimensions

Based on a linear finite element space,two symmetric finite volume schemes for eigenvalue problems in arbitrary dimensions are constructed and analyzed.Some relationships between the finite element method and the finite difference method are addressed,too.     

Fekete and Szeg problem in one and higher dimensions

In this paper, we establish the Fekete and Szeg inequality for a class of holomorphic functions in the unit disk, and then we extend this result to a class of holomorphic mappings on the unit ball in a complex Banach space or on the unit polydisk in C~n.     

FGT-injective dimensions of Π-coherent rings and almost excellent extension

We study, in this article, the FGT-injective dimensions of Π-coherent rings. If R is right Π-coherent, and TI (resp. TF) stands for the class of FGT-injective (resp. FGT-flat) R-modules (n ≥ 0), we show that the following are equivalent:

Hausdorff dimensions of some irregular sets associated with β-expansions

The Hausdorff dimensions of some refined irregular sets associated with β-expansions are determined for any β 1. More precisely, Hausdorff dimensions of the sets {x ∈ [0, 1) :lim inf(n→∞) S_n(x, β)/n= α_1, lim sup (n→∞) S_n(x, β)/n= α_2}, α_1, α_2≥0 are obtained completely, where S_n(x, β) =sum ε_k(x, β) from k=1 to n denotes the sum of the first n digits of the β-expansion of x. As an application, we present another concise proof of that the set of points x ∈ [0, 1) satisfying lim_(n→∞) S_n(x,β)/n does not exist is of full Hausdorff dimension.     

Euclidean and Lorentzian quantum gravity—lessons from two dimensions

No theory of four-dimensional quantum gravity exists as yet. In this situation the two-dimensional theory, which can be analyzed by conventional field-theoretical methods, can serve as a toy model for studying some aspects of quantum gravity. It represents one of the rare settings in a quantum-gravitational context where one can calculate quantities truly independent of any background geometry.We review recent progress in our understanding of 2d quantum gravity, and in particular the relation between the Euclidean and Lorentzian sectors of the quantum theory. We show that conventional 2d Euclidean quantum gravity can be obtained from Lorentzian quantum gravity by an analytic continuation only if we allow for spatial topology changes in the latter. Once this is done, one obtains a theory of quantum gravity where space-time is fractal: the intrinsic Hausdorff dimension of usual 2d Euclidean quantum gravity is four, and not two. However, certain aspects of quantum space-time remain two-dimensional, exemplified by the fact that its so-called spectral dimension is equal to two.     

Li–Yorke chaos in higher dimensions: a review

Li and Yorke not only introduced the term “chaos” along with a mathematically rigorous definition of what they meant by it, but also gave a condition for chaos in scalar difference equations, their equally famous “period three implies chaos” result. Generalizations of the Li and Yorke definition of chaos to difference equations in ? ⁿ are reviewed here as well as higher dimensional conditions ensuring its existence, specifically the “snap-back repeller” condition of Marotto and its counterpart for saddle points. In addition, further generalizations to mappings in Banach spaces and complete metric spaces are considered. These will be illustrated with various simple examples including an application to chaotic dynamics on the metric space (?  ⁿ , D) of fuzzy sets on the base space ? ⁿ .     

Positive recurrence for reflecting Brownian motion in?higher dimensions

Semimartingale reflecting Brownian motions (SRBMs) are diffusion processes with state space the d-dimensional nonnegative orthant, in the interior of which the processes evolve according to a Brownian motion, and that reflect against the boundary in a specified manner. A standard problem is to determine under what conditions the process is positive recurrent. Necessary and sufficient conditions for positive recurrence are easy to formulate in d=2, but not in d??3. Fluid paths are solutions of deterministic equations that correspond to the random equations of the SRBM. A standard result of Dupuis and Williams (in Ann. Probab. 22:680?C702, 1994) states that when every fluid path associated with the SRBM is attracted to the origin, the SRBM is positive recurrent. Employing this result, El Kharroubi et al. (in Stoch. Stoch. Rep. 68:229?C253, 2000; Math. Methods Oper. Res. 56:243?C258, 2002) gave sufficient conditions involving fluid paths for positive recurrence of SRBM in d=3. Here, we discuss two recent results regarding necessary conditions for positive recurrence of SRBM in d??3. Bramson et al. (in Ann. Appl. Probab. 20:753?C783, 2010) showed that the conditions in El Kharroubi et al. (Math. Methods Oper. Res. 56:243?C258, 2002) are, in fact, necessary in d=3. On the other hand, Bramson (in Ann. Appl. Probab., to appear, 2011) provided a family of positive recurrent SRBMs, in d??6, with linear fluid paths that diverge to infinity. The latter result shows in particular that the converse of the Dupuis?CWilliams result does not hold.     

A quadratic quasi-linear Klein–Gordon equation in two space dimensions

We consider the quasi-linear Klein–Gordon equations in two space dimensions $$\left(\partial_{t}^{2} - \Delta + 1\right) u=\mathcal{N} (u)$$ in ${(t, x) \in \mathbf{R} \times \mathbf{R}^{2}}$ with a quadratic nonlinearity ${\mathcal{N} (u)}$ , which is linear with respect to the second-order derivatives of unknown functions.     

A non-conforming domain decomposition for Raviart-Thomas vector fields in three dimensions

In this paper we are concerned with a domain decomposition method with nonmatching grids for Raviart-Thomas finite elements. In this method, the normal complement of the resulting approximation is not continuous across the interface. To handle such non-conformity, a new matching condition will be introduced. Such matching condition still     

Quadratic derivative nonlinear Schrödinger equations in two space dimensions

We study the global in time existence of small classical solutions to the nonlinear Schrödinger equation with quadratic interactions of derivative type in two space dimensions $\left\{\begin{array}{l@{\quad}l}i \partial _{t} u+\frac{1}{2}\Delta u=\mathcal{N}\left( \nabla u,\nabla u\right),&;t >0 ,\;x\in {\bf R}^{2},\\ u\left( 0,x\right) =u_{0} \left( x\right),&;x\in {\bf R}^{2}, \end{array}\right.\quad\quad\quad\quad\quad\quad (0.1)$ where the quadratic nonlinearity has the form ${\mathcal{N}( \nabla u,\nabla v) =\sum_{k,l=1,2}\lambda _{kl} (\partial _{k}u) ( \partial _{l}v) }We study the global in time existence of small classical solutions to the nonlinear Schr?dinger equation with quadratic interactions of derivative type in two space dimensions

$\left\{{l@{\quad}l}i \partial _{t} u+\frac{1}{2}\Delta u=\mathcal{N}\left( \nabla u,\nabla u\right),&t >0 ,\;x\in {\bf R}^{2},\\ u\left( 0,x\right) =u_{0} \left( x\right),&x\in {\bf R}^{2}, \right.\quad\quad\quad\quad\quad\quad (0.1)$\left\{\begin{array}{l@{\quad}l}i \partial _{t} u+\frac{1}{2}\Delta u=\mathcal{N}\left( \nabla u,\nabla u\right),&t >0 ,\;x\in {\bf R}^{2},\\ u\left( 0,x\right) =u_{0} \left( x\right),&x\in {\bf R}^{2}, \end{array}\right.\quad\quad\quad\quad\quad\quad (0.1)      20. Projective and injective dimensions in the category σ[M]      Seyed Shahin Mousavi  Mina Abbaszade 《代数通讯》2018,46(4):1560-1575