On the multifractal spectrum of weighted Birkhoff averages

<p style='text-indent:20px;'>In this paper, we study the topological spectrum of weighted Birk–hoff averages over aperiodic and irreducible subshifts of finite type. We show that for a uniformly continuous family of potentials, the spectrum is continuous and concave over its domain. In case of typical weights with respect to some ergodic quasi-Bernoulli measure, we determine the spectrum. Moreover, in case of full shift and under the assumption that the potentials depend only on the first coordinate, we show that our result is applicable for regular weights, like Möbius sequence.</p>

Continuity and Analyticity for the Generalized Benjamin–Ono Equation

This work mainly focuses on the continuity and analyticity for the generalized Benjamin–Ono (g-BO) equation. From the local well-posedness results for g-BO equation, we know that its solutions depend continuously on their initial data. In the present paper, we further show that such dependence is not uniformly continuous in Sobolev spaces Hs(R) with s>3/2. We also provide more information about the stability of the data-solution map, i.e., the solution map for g-BO equation is Hölder continuous in Hr-topology for all 0≤r<s with exponent α depending on s and r. Finally, applying the generalized Ovsyannikov type theorem and the basic properties of Sobolev–Gevrey spaces, we prove the Gevrey regularity and analyticity for the g-BO equation. In addition, by the symmetry of the spatial variable, we obtain a lower bound of the lifespan and the continuity of the data-to-solution map.

Approximation properties of multivariate exponential sampling series

In this paper, we generalize the family of exponential sampling series for functions of $n$ variables and study their pointwise and uniform convergence as well as the rate of convergence for the functions belonging to space of $\log$-uniformly continuous functions. Furthermore, we state and prove the generalized Mellin-Taylor's expansion of multivariate functions. Using this expansion we establish pointwise asymptotic behaviour of the series by means of Voronovskaja type theorem.

Approximations of Symmetric Functions on Banach Spaces with Symmetric Bases

This paper is devoted to studying approximations of symmetric continuous functions by symmetric analytic functions on a Banach space X with a symmetric basis. We obtain some positive results for the case when X admits a separating polynomial using a symmetrization operator. However, even in this case, there is a counter-example because the symmetrization operator is well defined only on a narrow, proper subspace of the space of analytic functions on X. For X=c0, we introduce ε-slice G-analytic functions that have a behavior similar to G-analytic functions at points x∈c0 such that all coordinates of x are greater than ε, and we prove a theorem on approximations of uniformly continuous functions on c0 by ε-slice G-analytic functions.

Generalized John Gromov hyperbolic domains and extensions of maps

Let $\Omega \subset \mathbb{R}^n$ be a Gromov hyperbolic, $\varphi$-length John domain. We show that there is a uniformly continuous identification between the inner boundary of $\Omega$ and the Gromov boundary endowed with a visual metric, By using this result, we prove the boundary continuity not only for quasiconformal homeomorphisms, but also for more generally rough quasi-isometries between the domains equipped with the quasihyperbolic metrics.

Index boundedness and uniform connectedness of space of the G-permutation degree

<p>In this paper the properties of space of the G-permutation degree, like: weight, uniform connectedness and index boundedness are studied. It is proved that:</p><p>(1) If (X, U) is a uniform space, then the mapping π s n, G : (X n , U n ) → (SP n GX, SP n GU) is uniformly continuous and uniformly open, moreover w (U) = w (SP n GU);</p><p>(2) If the mapping f : (X, U) → (Y, V) is a uniformly continuous (open), then the mapping SP n Gf : (SP n GX, SP n GU) → (SP n GY, SP n GV) is also uniformly continuous (open);</p><p>(3) If the uniform space (X, U) is uniformly connected, then the uniform space (SP n GX, SP n GU) is also uniformly connected.</p>

Algebras of entire functions containing functions of unbounded type on a Banach space

In this paper, we consider algebras of entire analytic functions which are bounded on a prescribed family of balls in a Banach space. We investigate the structures of such algebras and describe their spectra in terms of spectra of algebras of uniformly continuous analytic functions. Some partial examples are considered. In particular, we have complete descriptions of the spectra for the case of Tsirelson space and for $c_0$.

Computing with continuous objects: a uniform co-inductive approach

A uniform approach to computing with infinite objects like real numbers, tuples of these, compacts sets and uniformly continuous maps is presented. In the work of Berger, it was shown how to extract certified algorithms working with the signed digit representation from constructive proofs. Berger and the present author generalised this approach to complete metric spaces and showed how to deal with compact sets. Here, we unify this work and lay the foundations for doing a similar thing for the much more comprehensive class of compact Hausdorff spaces occurring in applications. The approach is of the same computational power as Weihrauch’s Type-Two Theory of Effectivity.

Singularities of solutions of time dependent Hamilton-Jacobi equations. Applications to Riemannian geometry

If $U:[0,+\infty [\times M$ U : [ 0 , + ∞ [ × M is a uniformly continuous viscosity solution of the evolution Hamilton-Jacobi equation $$ \partial _{t}U+ H(x,\partial _{x}U)=0, $$ ∂ t U + H ( x , ∂ x U ) = 0 , where $M$ M is a not necessarily compact manifold, and $H$ H is a Tonelli Hamiltonian, we prove the set $\Sigma (U)$ Σ ( U ) , of points in $]0,+\infty [\times M$ ] 0 , + ∞ [ × M where $U$ U is not differentiable, is locally contractible. Moreover, we study the homotopy type of $\Sigma (U)$ Σ ( U ) . We also give an application to the singularities of the distance function to a closed subset of a complete Riemannian manifold.