Weakly dependent chains with infinite memory

We prove the existence of a weakly dependent strictly stationary solution of the equation _Xt=F(_Xt−1,_Xt−2,_Xt−3,…;_ξt)$X_t=F(X_{t-1},X_{t-2},X_{t-3},\dots ;{\xi }_t)$ called a chain with infinite memory. Here the innovations  _ξt${\xi }_t$ constitute an independent and identically distributed sequence of random variables. The function F$F$ takes values in some Banach space and satisfies a Lipschitz-type condition. We also study the interplay between the existence of moments, the rate of decay of the Lipschitz coefficients of the function F$F$ and the weak dependence properties. From these weak dependence properties, we derive strong laws of large number, a central limit theorem and a strong invariance principle.     

Aluthge Transforms of Complex Symmetric Operators

If denotes the polar decomposition of a bounded linear operator T, then the Aluthge transform of T is defined to be the operator . In this note we study the relationship between the Aluthge transform and the class of complex symmetric operators (T iscomplex symmetric if there exists a conjugate-linear, isometric involution so that T = CT*C). In this note we prove that: (1) the Aluthge transform of a complex symmetric operator is complex symmetric, (2) if T is complex symmetric, then and are unitarily equivalent, (3) if T is complex symmetric, then if and only if T is normal, (4) if and only if T ² = 0, and (5) every operator which satisfies T ² = 0 is necessarily complex symmetric. This work partially supported by National Science Foundation Grant DMS 0638789.     

Evaluation of recurrence relationships and numerical solutions for finite rotation matrix elements by using auxiliary functions

In this article, we derived several new recurrence relations of the rotation matrix elements by using Gauss’ recurrence formulas for hypergeometric functions and auxiliary functions , and . The aim of this contribution is to obtain general algorithm to compute the rotation matrix elements, paying attention to the use recurrence relationships of the auxiliary functions that allow the treatment of the functions with high angular momentum quantum numbers.     

Site-Specific m6A Erasing via Conditionally Stabilized CRISPR-Cas13b Editor

N6-methyladenosine (m⁶A) on RNAs plays an important role in regulating various biological processes and CRIPSR technology has been employed for programmable m⁶A editing. However, the bulky size of CRISPR protein and constitutively expressed CRISPR/RNA editing enzymes can interfere with the native function of target RNAs and cells. Herein, we reported a conditional m⁶A editing platform (FKBP*-dCas13b-ALK) based on a ligand stabilized dCas13 editor. The inducible expression of this m⁶A editing system was achieved by adding or removing the Shield-1 molecule. We further demonstrated that the targeted recruitment of dCas13b-m⁶A eraser fusion protein and site-specific m⁶A erasing were achieved under the control of Shield-1. Moreover, the release and degradation of dCas13b fusion protein occurred faster than the restoration of m⁶A on the target RNAs after Shield-1 removal, which provides an ideal opportunity to study the m⁶A function with minimal steric interference from bulky dCas13b fusion protein.     

Errata to: “Vector-valued Modulation Spaces and Localization Operators with Operator-valued Symbols”

In our article “Vector-valued Modulation Spaces and Localization Operators with Operator-valued Symbols”, Integr. equ. oper. theory 59 (1) (2007), 99–128, Example 4.4 is false. We present a corrected example.      

A Gamma-convergence approach to the Cahn–Hilliard equation

We study the asymptotic dynamics of the Cahn–Hilliard equation via the “Gamma-convergence” of gradient flows scheme initiated by Sandier and Serfaty. This gives rise to an H ¹-version of a conjecture by De Giorgi, namely, the slope of the Allen–Cahn functional with respect to the H ⁻¹-structure Gamma-converges to a homogeneous Sobolev norm of the scalar mean curvature of the limiting interface. We confirm this conjecture in the case of constant multiplicity of the limiting interface. Finally, under suitable conditions for which the conjecture is true, we prove that the limiting dynamics for the Cahn–Hilliard equation is motion by Mullins–Sekerka law. Partially supported by a Vietnam Education Foundation graduate fellowship.     

On the regularity of the pressure field of Brenier’s weak solutions to incompressible Euler equations

In this paper we improve the regularity in time of the gradient of the pressure field arising in Brenier’s variational weak solutions (Comm Pure Appl Math 52:411–452, 1999) to incompressible Euler equations. This improvement is necessary to obtain that the pressure field is not only a measure, but a function in . In turn, this is a fundamental ingredient in the analysis made by Ambrosio and Figalli (2007, preprint) of the necessary and sufficient optimality conditions for the variational problem by Brenier (J Am Mat Soc 2:225–255, 1989; Comm Pure Appl Math 52:411–452, 1999).     

Characterizing the round sphere by mean distance

We discuss the measure theoretic metric invariants extent, rendezvous number and mean distance of a general compact metric space X and relate these to classical metric invariants such as diameter and radius. In the final section we focus attention to the category of Riemannian manifolds. The main result of this paper is Theorem 4 stating that the round sphere of constant curvature 1 has maximal mean distance among Riemannian n-manifolds with Ricci curvature Ric?n−1, and that such a manifold is diffeomorphic to a sphere if the mean distance is close to .     

On Attractors for Multivalued Semigroups Defined by Generalized Semiflows

In this paper we extend results from Semigroup Theory on existence and characterization of attractors in order to include multivalued semigroups T(t) defined by generalized semiflows . In particular we show that, if is continuous, possesses a Lyapunov function, and has a global attractor which is maximal compact invariant, then  =  W ^u (Z()), where Z() is the stationary solutions set and W ^u (Z()) is the unstable set of Z(). We introduce the -attractor concept which does not enjoy any uniformity on time of attraction and we prove, under suitable conditions, that the global -attractor is the set of asymptotic states described by Z(). Jacson Simsen is supported by CAPES-Brazil.